
a sequence of contractions
\(\tau _s : \Gamma ({\Sigma }_{s_0}) \rightarrow \Gamma \) , constant in \(s \in S_\Gamma \) ;

for every node \( \lbrace  w_i^\pm \rbrace \)  collapsed under \(\tau _s\) , a pair
of local coordinates \(\kappa ^\pm _i: W^\pm _i \rightarrow \item for every component \) s0,i\( of \) s0\(, areal number \) s > 0\( converging to \) 0\( as \) s s0\( and maps\)\(\phi _{i,s}: \Sigma _{s_0,i} - \cup _{w_k^\pm \in \Sigma _{s_0,i},\tau _s(w_k^\pm ) = \emptyset } (\kappa _k^\pm )^{-1}(B_{\epsilon _s}) \rightarrow {\Sigma }_{s,\tau _s(i)}\)\(\) 
such that

for any \(s\) , the images of the maps \(\phi _{i,s}\) 
cover \({\Sigma }_{s}\) ;

for any nodal point \(w_i^\pm \)  of \({\Sigma }_s\)  joining
components \(\Sigma _{s,i^\pm (k)}\) , there exists a constant \(\lambda _s\in *\)  such that \((\kappa _k^+ \circ \phi _{s,i^+(k)}^{-1} \circ \phi _{s,i^-(k)} \circ (\kappa _k^-)^{-1})(z) = \lambda _s z \)  where
defined, and \(\lambda _s \rightarrow 0\)  as \(s \rightarrow s_0\) .

for any \(z \in \Sigma _{s_0,i}\)  in the complement of the
\(W_{k,s}^\pm \) , \(\lim _{s \rightarrow s_0}(\phi _{i,s}(z)) = z\) ;

\(\phi _{i,s}^* j_{\Sigma _{s,\tau _s(i)}}\)  converges to \(j_{\Sigma _{s_0,i}}\)  uniformly in all derivatives on compact sets;

if \(z_i\)  is contained in \(\Sigma _{s_0,k}\) , then \(z_i = \lim _{s\rightarrow s_0} \phi _{s,k}^{-1}(z_{i,s})\) .

\(\phi _{i,s}^* u_s\)  converges to \(u_{s_0}\)  uniformly in all
derivatives on compact sets.

Remark 3.1.2 It follows from the assumption that \(u_S: \Sigma _S \rightarrow X\) 
is continuous that the homology class \(u_{s,*} [\Sigma _s]\)  is locally
constant in \(s \in S\) . Indeed continuity implies that for \(s\) 
sufficiently close to \(s_0\) , \(u_S\)  is homotopic to a map of the form
\(v_S \circ \gamma _S\)  where \(\gamma _s: \Sigma _S \rightarrow \Sigma _{s_0}\)  is a
map to the central fiber \(\Sigma _{s_0}\)  which collapses the gluing
regions to the node. Since each \(\gamma _s = \gamma _S | \Sigma _s\)  maps
\([\Sigma _s]\)  to \([\Sigma _{s_0}]\) , the claim follows.

In particular, taking \(S\)  to be the topological space given as the
closure of the set \(S^*\)  of rational numbers of the form \(1/i, i \in \mathbb {Z}_{> 0}\) , we say that a sequence of holomorphic maps \(u_i: \Sigma _i\rightarrow X\)  Gromov converges if it extends to a continuous family
over \(S\) . To state the Gromov compactness theorem, recall that the
energy of a map \(u: \Sigma \rightarrow X\)  is
\( E(u)=\frac{1}{2}\int |du|^2 .\) 
Theorem 3.1.3 (Gromov compactness) 
Let \(X,\omega ,J\)  be as above. Any sequence \(u_i: \Sigma _i \rightarrow X\)  of
stable holomorphic maps with bounded energy has a Gromov convergent
subsequence. Furthermore, the limit is unique.

For references and discussion, see for example [1]}. The definition of Gromov convergence passes naturally
to equivalence classes of stable maps. A subset \(C\)  of
\(\overline{M}_{g,n}(X,d)\)  is Gromov closed if any sequence in \(C\)  has
a limit point in \(C\) , and Gromov open if its complement is
closed. The Gromov open sets form a topology for which any convergent
sequence is Gromov convergent, by an argument using [2]}. Furthermore, any convergent sequence has a unique
limit. Gromov compactness implies that for any \(E> 0\) , the union of
\(\overline{M}_{g,n}(X,d)\)  over \(d \in H_2(X,\mathbb {Z})\)  with \((d, [\omega ]) < E\)  is
a compact, Hausdorff space.
Definition 3.1.4 
Let \(X,\omega ,J\)  be as above.
A stratified-smooth family of nodal \(J\) -holomorphic maps over a
space \(S\)  is a pair \(({\Sigma }_S,{u}_S)\)  of a stratified-smooth
family of nodal curves \({\Sigma }_S \rightarrow S\)  together with a
continuous map \({u}_S: {\Sigma }_S \rightarrow X\)  such that the
restriction \({u}_s\)  of \({u}\)  to any fiber \({\Sigma }_s\)  is
holomorphic, and the restriction of \({u}_S\)  to any stratum
\({\Sigma }_\Gamma \)  is smooth.
A stratified-smooth deformation of a stable \(J\) -holomorphic map
\(({\Sigma }, {u})\)  is a germ of a stratified-smooth family
\(({\Sigma }_S, {u}_S)\)  together with an isomorphism
of nodal maps \(\iota : {\Sigma }_0 \rightarrow {\Sigma }\)  such that
\(\iota ^* {u} = {u}_0\) .
A deformation \(({\Sigma }_S, {u}_S,\iota )\)  of
\(({\Sigma },{u})\)  is versal if any other (germ of) family
of marked, nodal curves \(({\Sigma }^{\prime },{\Sigma }_0) \rightarrow (S^{\prime },0)\)  is
induced from a map \(\psi : S^{\prime } \rightarrow S\)  in the sense that there exists
an isomorphism \(\phi : {\Sigma }^{\prime } \rightarrow {\Sigma } \times _S S^{\prime }\)  in a
neighborhood of the central fiber \( {\Sigma }_0\) , and \({u}^{\prime }\)  is
obtained by composing projection on the first factor with \({u}\) .
A versal deformation is universal if the map \(\phi \)  above is
the unique map inducing the identity on \({\Sigma }_0\) .

Smooth universal deformations of regular stable maps of fixed
combinatorial type.
Let \({u} :{\Sigma } \rightarrow X\)  be a stable map. For \(p >2\)  define
a fiber bundle \(\mathcal {E}\rightarrow \mathcal {B}\)  by
\( \mathcal {B}= J({\Sigma }) \times \operatorname{Map}({\Sigma },X)_{1,p} , \quad \mathcal {E}_{\zeta ,{u}} = \Omega ^{0,1}({\Sigma }, {u}^* TX)_{0,p},\) 
where the latter is the space of \((0,1)\) -forms with respect to the
pair \(({j}(\zeta ),J)\) . Consider the Cauchy-Riemann section,
\( \overline{\partial }:\mathcal {B}\rightarrow \mathcal {E}, \ \ \ (j,{u}) \mapsto \overline{\partial }_j {u},\ \ \overline{\partial }_j {u} = {{2}}( {\operatorname{d}}{u} \circ {j}(\zeta ) - J_u \circ {\operatorname{d}}{u}) .\) 
Let
\( \operatorname{ev}: \mathcal {B}\rightarrow X^{2m} , \ \ \ {u} \mapsto ({u}(w_1^-), {u}(w_1^+), \ldots ,{u}(w_m^-), {u}(w_m^+) ) \) 
denote the map evaluating at the nodal points. The space of stable
maps of type \(\Gamma \)  is given as \((\overline{\partial },\operatorname{ev})^{-1}(0)\) . To obtain a
Fredholm map, we quotient by diffeomorphisms of \(\Sigma \) , or
equivalently, restrict to a minimal versal deformation \({\Sigma }_S\rightarrow S\)  of \({\Sigma }\)  of fixed type. This means that for each
\(\zeta \in \operatorname{Def}_\Gamma ({\Sigma })\)  near 0 we have a complex
structure \({j}(\zeta )\)  on \({\Sigma }\) , which we may assume
agrees with \({j} = {j}(0)\)  near the nodes. Then the
Cauchy-Riemann section induces a map
\( \operatorname{Def}_\Gamma (\Sigma ) \times \Omega ^0(\Sigma , u^* TX) \rightarrow \mathcal {E}.\) 
Linearizing the Cauchy-Riemann section, together with the differences
at the nodes, gives rise to a Fredholm operator
\(\tilde{D}_{{u}} : \operatorname{Def}_\Gamma ({\Sigma }) \times \Omega ^0({\Sigma }, {u}^* TX) \rightarrow \Omega ^{0,1}({\Sigma },{u}^* TX) \oplus \bigoplus _{i=1}^m{u}(w_i^\pm )^* TX   \tilde{D}_{{u}} ({\zeta },{\xi }) :=\left( \pi ^{0,1}_{{\Sigma }} ( \nabla {\xi } - {{2}}J({u}) {\operatorname{d}}{u} Dj({\zeta }) - {{2}}J_{{u}} (\nabla _{{\xi }} J )_{{u}}\partial {u})), ( {\xi }(w_i^+)- {\xi }(w_i^-) )_{i=1}^m\right)\) 
given by the linearized Cauchy-Riemann operator on each component, and
the difference of the values of the section at the nodes
\(w^\pm _1,\ldots , w^\pm _m\) . The map \({u} =({\Sigma },{u},{z})\)  is regular if \(\tilde{D}_{{u}}\) 
is surjective. This is independent of the choice of representatives
\(j(\zeta )\) : any two such choices \({j}^{\prime }(\zeta ),{j}(\zeta )\)  are
related by a diffeomorphism of \({\Sigma }\) . The space of
infinitesimal deformations of \({u}\)  of fixed type is
\( \operatorname{Def}_\Gamma ({u}) =  \operatorname{ker}(\tilde{D}_{{u}})/ \operatorname{aut} ({\Sigma }) .\) 
The space of infinitesimal deformations of \({u}\) is
\( \operatorname{Def}({u}) = \operatorname{Def}_\Gamma ({u}) \oplus \bigoplus _{i=1}^mT_{w_j^+} {\Sigma } \otimes T_{w_j^-} {\Sigma } \) 
where \(\Gamma \)  is the type of \({u}\) .
Theorem 3.2.1  Let \(X,\omega ,J\)  be as above.
A regular marked nodal \(J\) -holomorphic map \({u} =({\Sigma },{u},{z})\)  admits a strongly universal deformation
\(({\Sigma }_S,{u}_S,{z}_S)\)  with parameter space \(S \subset \operatorname{Def}_\Gamma ({u})\)  of fixed type if and only if \({u}\)  is stable.

Let \(({\Sigma },{u})\)  be a stable map to \(X\)  and
\({\Sigma }_S \rightarrow S \subset \operatorname{Def}_\Gamma ({\Sigma })\)  a minimal
versal deformation of \({\Sigma }\)  of fixed type constructed in
(REF ). We may write any map \(C^0\) -close to \({u}\)  as
\(\exp _{{u}} ({\xi })\)  for some \({\xi } \in \Omega ^0({\Sigma }, {u}^* TX)\) . Let \(\Psi _{{u}}({\xi }):{u}^* TX \rightarrow \exp _{{u}}({\xi })^* TX\)  denote parallel
transport along geodesics with respect to the Hermitian connection \(\tilde{\nabla } = \nabla - {{2}}J(\nabla J) ;\)  here \(\nabla \)  is the
Levi-Civita connection, see [2]}. This defines an
isomorphism
\( \Psi _{{u}}({\xi })^{-1} : \Omega ^{0,1}_{{j}}( {\Sigma },\exp _{{u}}({\xi })^* TX) \rightarrow \Omega ^{0,1}_{{j}}({\Sigma }, {u}^* TX).\) 
where subscript \(j\)  denotes the space of \(0,1\) -forms taken with
respect to the complex structure \(j\)  on \({\Sigma }\) . There is an
isomorphism of \(\Omega ^{0,1}_{{j}(\zeta )}({\Sigma }, {u}^*TX)\)  with \(\Omega ^{0,1}_{{j}}({\Sigma },{u}^*TX)\)  given by
composing the inclusion
\(\Omega ^{0,1}_{{j}(\zeta )}({\Sigma },{u}^* TX) \rightarrow \Omega ^1({\Sigma },{u}^* TX)_ \Omega ^1({\Sigma }; {u}^* TX) \otimes _\mathbb {R}\) 
with the projection \( \Omega ^1({\Sigma }; {u}^* TX) \otimes _\mathbb {R}\Omega ^{0,1}_{{j}}({\Sigma },{u}^* TX) .\)  We denote by
\( \Psi _j(\zeta ): \Omega ^{0,1}_{{j}}({\Sigma }, {u}^* TX) \rightarrow \Omega ^{0,1}_{{j}(\zeta )}({\Sigma },{u}^* TX) \) 
the resulting map; one can think of this as a connection over the
space of complex structures on \({\Sigma }\)  on the bundle whose fiber
is the space of \(0,1\) -forms with respect to \(j(\zeta )\) . By composing
\(\Psi _{{u}}({\xi })^{-1}\)  and \(\Psi _j(\zeta )^{-1}\)  we
obtain an identification
\( \Psi _{j,{u}}(\zeta ,\xi )^{-1} :\Omega ^{0,1}_{{j}(\zeta )}({\Sigma },\exp _{{u}}({\xi })^* TX) \rightarrow \Omega ^{0,1}_{{j}}({\Sigma },{u}^*TX) .\) 
Define
\( \mathcal {F}_{{u}}: \operatorname{Def}_\Gamma ({\Sigma }) \times \Omega ^0({\Sigma },{u}^* TX) \rightarrow \Omega ^{0,1}_{{j}}({\Sigma },{u}^* TX) \) 
\( (\zeta ,{\xi }) \mapsto \Psi _{j,{u}}(\zeta ,{\xi })^{-1}(\overline{\partial }_{{j}(\zeta )}(\exp _{{u}}({\xi }))) .\) 
The operator \(\tilde{D}_{{u}}\)  is the linearization of \(\mathcal {F}_{{u}}\) .
The implicit function theorem implies that if \({u}\)  is regular then
the zero set of \(\mathcal {F}_{{u}}\)  is modelled locally on a neighborhood
of 0 in \( \operatorname{ker}(\tilde{D}_{{u}})\) . Furthermore, by elliptic
regularity the zero set consists entirely of smooth \(J\) -holomorphic
maps [2]}. Thus we obtain a smooth family of
stable maps in a neighborhood of 0 in \( \operatorname{ker}(\tilde{D}_{{u}})\) . The
action of \( \operatorname{Aut} ({u})\)  on the space of stable maps with domain
\({\Sigma }\)  induces an inclusion of the Lie algebra \( \operatorname{aut} ({u})\) 
into \( \operatorname{ker}(\tilde{D}_{{u}})\) . Restricting to \(\operatorname{Def}_\Gamma ({u})\) ,
identified with a complement of \( \operatorname{aut} ({\Sigma })\)  (that is, a slice
for the \( \operatorname{Aut} ({u})\)  action) gives a family \(({\Sigma }_S,{u}_S) \rightarrow S \subset \operatorname{Def}_\Gamma ({u})\)  of fixed type. The
family \(({\Sigma }_S,{u}_S)\) , together with the canonical
identification \(\iota \)  of the central fiber with \({\Sigma }\) , is a
universal smooth deformation of fixed type. Indeed, another smooth
family \(({\Sigma }_{S^{\prime }},{u}_{S^{\prime }})\)  over a base \(S^{\prime }\)  is in
particular a deformation of the underlying curve. After shrinking
\(S^{\prime }\) , each fiber of \(({\Sigma }_{s^{\prime }}, {u}_{s^{\prime }}^{\prime })\)  corresponds to
a zero of \(\mathcal {F}_{{u}}\) , and so lies in the image of the map given by
the implicit function theorem. The uniqueness part of the implicit
function theorem gives a smooth map \(\psi : S^{\prime } \rightarrow \operatorname{Def}_\Gamma ({u})\) 
and an identification \({\Sigma }_{S^{\prime }} \rightarrow \psi ^* {\Sigma }_S\) .
Any two such maps inducing the same map on the central fiber are close
in a neighborhood of the central fiber. Since the automorphism groups
of the central fibre are discrete, any automorphism group is
discrete. Thus any two such automorphisms defined in a neighborhood of
the central fiber, and equal on the central fiber must be equal in a
neighborhood of the central fiber. This shows that the identification
is unique, so that the deformation given by the gluing construction is
universal.
If \({u}\)  is not stable, then it has no universal deformation since
the identification with the central fiber is unique only up to a
continuous family of automorphisms.
Let \(M_{g,n,\Gamma }^{{\operatorname{reg}}}(X,d)\)  denote the moduli space of regular
stable maps of combinatorial type \(\Gamma \) . A family \( {u}_S\)  over
\(S \subset \operatorname{Def}_\Gamma ({u}) \)  induces a map
\( S \rightarrow M_{g,n,\Gamma }^{{\operatorname{reg}}}(X,d), \ s \mapsto [{u}_s]\) 
where \([{u}_s]\)  denotes the isomorphism class of \({u}_s:{\Sigma }_s \rightarrow X\) .
Theorem 3.2.2 For any \(g,n,d\)  and combinatorial type \(\Gamma \) 
with \(m\)  nodes, \(M_{g,n,\Gamma }^{{\operatorname{reg}}}(X,d)\)  has the structure of a
smooth orbifold of dimension \((1-g)( \dim (X) - 6) + 2 (c_1 (TX),d) -2m + 2n\) , with tangent space at \([{u}]\)  isomorphic to
\(\operatorname{Def}_\Gamma ({u})\) .

By Theorem REF , the maps (REF ) for families
giving universal deformations are homeomorphisms onto their image and
provide compatible charts. The dimension formula follows from
Riemann-Roch: The index of \(\tilde{D}_{{u}}\)  may be deformed to a
complex linear operator by homotoping the zero-th order terms (which
define a compact operator) to zero.

Constructing stratified-smooth deformations of varying type
The main result of this section is Theorem REF , which is
probably well-known, cf. [5]}, [6]}, but for which
we could not find an explicit reference. The theorem itself will not
be used, but the estimates involved in the proof will be needed later
for the corresponding result for vortices. The proof uses a gluing
construction for holomorphic maps, which produces from a smooth family
of holomorphic maps of fixed type, a stratified-smooth family of maps
of varying type.
Step 1: Approximate Solution
Definition 3.3.1 Let \({\Sigma }\)  be a compact, complex nodal curve.
A gluing datum for \({\Sigma }\)  consists of

a collection of gluing parameters \({\delta } =(\delta _1,\ldots ,\delta _m)\)  in the bundle \({I}\)  of
(REF );

local coordinates \(\kappa _j^\pm \)  near the nodes \(w_j^\pm \)  for
\(j=1,\ldots ,m\) ;

a parameter \(\rho \)  which describes the width of the annulus on
which the gluing of maps is performed;

a gluing profile \(\varphi \) , see Definition REF ;

a smooth cutoff function
\(\alpha : [0,1 ], \ \ \ \alpha (z) = \left\lbrace \begin{array}{ll} 0 & |z| \le 1 \\1 & |z| \ge 2 \end{array} \right\rbrace  .\) 


We first treat the case that \(\varphi \)  is the standard gluing profile.
Let a gluing datum be given, and \( {\Sigma }^{{\delta }}\)  denote
the glued curve from (REF ). Let \({u}: {\Sigma } \rightarrow X\) 
be a holomorphic map. Near each node \(w_k\)  let \(i^\pm (k)\)  denote the
components on either side of \(w_k\) . In the neighborhoods \(U_k^\pm \) 
(assuming they have been chosen sufficiently small) define maps
\( \xi _k^\pm : U_k^\pm \rightarrow T_{x_k} X, \ \ \ u_{i^\pm (k)}(z) =\exp _{x_k}(\xi _k^\pm (z)) \) 
where \(x_k = u(w_k)\)  and \(\exp _{x_k}: T_{x_k} X \rightarrow X\)  denotes
geodesic exponentiation. Given a holomorphic map \({u}:{\Sigma }\rightarrow X\) , and a gluing datum
\(({\delta },{\kappa },\rho ,\varphi ,\alpha )\)  define the pre-glued map by interpolating between the maps on the various
components using the given cutoff function and local coordinates:
\({u}^{{\delta }} = {u}(z)\)  for \(z \notin \cup _k U_k^\pm \)  and
otherwise
\( {u}^{{\delta }} (z) = \exp _{x_k}( \alpha (\kappa _k^\pm ( z) / \rho |\delta _k|^{1/2} ) \xi _k^\pm (z) ) \quad z \in U_k^\pm .\) 
Remark 3.3.2 
The same formula but with domain \({\Sigma }\)  (not the glued curve)
defines an intermediate map \( {u}_0^{{\delta }}:{\Sigma } \rightarrow X \)  which is constant near the nodes. The right
inverse of \(\tilde{D}_{{u}_0^{{\delta }}}\)  will be used in the
gluing construction.

First we estimate the failure of
\({u}^{{\delta }}\)  to satisfy the Cauchy-Riemann equation. Define
on \({\Sigma }^{{\delta }}\)  the \(C^0\) -metric \(g\)  by the
identification
\(  {\Sigma }^{{\delta }} = {\Sigma } - \bigcup _{k,\pm }\kappa _k^\pm (B_{|\delta _k|^{1/2}}(0)) / \left( \kappa ^+_k(\partial B_{|\delta _k|^{1/2}}(0))\sim \kappa ^-_k(\partial B_{|\delta _k|^{1/2}}(0)) \right)\) 
using a Kähler metric on \(\Sigma \) , see Figure REF .
<FIGURE>The generalized Sobolev spaces \(W^{l,p}\)  with respect to
this metric are defined for \(p \ge 1\)  and integers \(l \in \lbrace 0,1\rbrace \) ,
see [7]} or [8]}. For any vector bundle \(E\)  we
denote by \(\Omega ({\Sigma }^{{\delta }}, E)_{l,p,{\delta }}\)  the
space of \(W^{l,p}\)  forms with values in \(E\) . If \(p = \infty \)  the norm
is independent of \({\delta }\)  and we drop it from the notation. Let
\(\Vert \cdot \Vert _{k,p,{\delta }}\)  denote the Sobolev
\(W^{k,p}\) -norm on \(\Omega ^0({u}^* TX)\)  defined using the
\({\delta }\) -dependent metric (REF ).
Proposition 3.3.3 
Suppose that \({u}: {\Sigma } \rightarrow X\)  is a stable map, and
\({u}^{{\delta }}: {\Sigma }^{{\delta }} \rightarrow X\)  is the
pre-glued map defined in (REF ), defined for \({\delta }\) 
sufficiently small. There is a constant \(c\)  and an \(\epsilon > 0\)  such
that if
\( \Vert {\delta } \Vert < \epsilon ,\rho > 1/\epsilon ,\)  and \( |\delta _k|^2 \rho < \epsilon ,\ \ \ k= 1,\ldots , m \) 
then
\(\Vert \overline{\partial } {u}^{{\delta }} \Vert _{0,p,{\delta }}^p\le c \sum _{k=1}^m (|\delta _k|^{1/2} \rho )^{2} .\) 

Compare with McDuff-Salamon [2]}. The error term
\(\overline{\partial } {u}({\delta })\)  can be estimated by terms of two
types; those involving derivatives of the cutoff functions and those
involving derivatives of the map \(\xi _k\) .
The derivative of \(\exp _{x_k}\)  is approximately the identity near the
node. The derivatives of \(\alpha \)  grow like \( 1/ \rho |\delta _k|^{1/2}\) , while the norm of \(\xi _k^\pm \)  is bounded by a
constant times \(\rho |\delta _k |^{1/2}\)  on the gluing region. The
term involving the derivatives of \(\alpha \)  is bounded and supported
on a region of area less than \( \pi \rho ^2 |\delta _k|\)  for each
node. The derivatives of \(\xi _k^\pm \)  are also uniformly bounded, and
the area bound gives the required estimate.
Let \({\Sigma }_S \rightarrow S\)  with \(S \subset \operatorname{Def}_\Gamma ({\Sigma })\) 
be a family giving a minimal versal deformation of \({\Sigma }\)  of
fixed type, and \({\Sigma }_{S_{{\delta }}({\delta })} \rightarrow S_{{\delta }} \subset \operatorname{Def}({\Sigma }^{{\delta }})\)  a family
giving a minimal versal deformation of \({\Sigma }^{{\delta }}\) .
The gluing construction (REF ) applied to the family
\({\Sigma }_S\)  produces a map
\( \operatorname{Def}_\Gamma ({\Sigma }) \rightarrow J({\Sigma }^{{\delta }}), \quad \zeta \mapsto j^{{\delta }}(\zeta )\) 
which maps any deformation of the original curve to the corresponding
deformation of the glued curve. In other words, any variation of
complex structure on \({\Sigma }\)  of fixed type induces a variation
of complex structure on \({\Sigma }^{{\delta }}\) . Similarly, for
any \(\xi \in \Omega ^0({\Sigma }, {u}^* TX)\)  we obtain an element
\(\xi ^{{\delta }} \in \Omega ^0({\Sigma }^{{\delta }}, {u}^*TX)\) .
Proposition 3.3.4 
Suppose that \({u}, {u}^{{\delta }}\)  are as above, and
\((\zeta ,\xi ) \in \operatorname{Def}_\Gamma (u)\) . There is a constant \(c\) 
and an \(\epsilon > 0\)  such that if
\( \Vert {\delta } \Vert < \epsilon , \rho > 1/\epsilon \) , \(\Vert \zeta \Vert +\Vert {\xi } \Vert _{1,p} \le \epsilon \) , and \( |\delta _k|^2 \rho < \epsilon \)  for
\( k= 1,\ldots , m \)  then
\(\Vert \overline{\partial }_{{j}^{{\delta }}(\zeta )}\exp _{{u}^{{\delta }}}({\xi }^{{\delta }})\Vert _{0,p,{\delta }}^p \le c \sum _{k=1}^m (|\delta _k|^{1/2}\rho )^{2} .\) 

Step 2: Uniformly bounded right inverse
We wish to show that the map in Proposition REF  can be
corrected to obtain a holomorphic map. Define
\( \mathcal {F}_{{u}}^{{\delta }}:\operatorname{Def}_\Gamma ({\Sigma })\times \Omega ^0({\Sigma }^{{\delta }},{u}^{{\delta },*} TX)\rightarrow \Omega ^{0,1}({\Sigma }^{{\delta }},{u}^{{\delta },*}TX) \) 
\( (\zeta ,\xi ) \mapsto \Psi _{j,{u}^{{\delta }}}(\zeta ,\xi )^{-1}(\overline{\partial }_{{j}^{{\delta }}(\zeta )}(\exp _{{u}^{{\delta }}}(\xi ))) .\) 
Here the operator \(\Psi _{j,{u}^\delta }\)  is as in
(REF ). Let \(\tilde{D}_{{u}}^{{\delta }}(\xi )\)  be the
associated linear operator, that is, the linearization of (REF )
at \(\xi \) . This operator naturally extends to a map from Sobolev
\(1,p\) -completion of the second factor of the domain to the
\(0,p\) -completion of the codomain. We denote by
\(\tilde{D}_{{u}}^{{\delta }} := \tilde{D}_{{u}}^{{\delta }}(0)\) .
We will construct an approximate inverse
\(T_{{\delta }} :\Omega ^{0,1}({\Sigma }^{{\delta }},{u}^{{\delta },*} TX ) \rightarrow \operatorname{Def}_\Gamma ({\Sigma }) \oplus \Omega ^0({\Sigma }^{{\delta }},{u}^{{\delta },*}TX)\) 
to \(\tilde{D}_{{u}}^{{\delta }}\) . The
construction depends on a carefully chosen cutoff function:
Lemma 3.3.5  [2]}
For any \(\delta > 0, \rho > 1\)  there exists a function
\(\beta _{\rho ,\delta } : \mathbb {R}^2 \rightarrow [0,1]\)  that satisfies
\( \beta _{\rho ,\delta }(z) = {\left\lbrace \begin{array}{ll} 0 & | z| \le \sqrt{\delta /\rho } \\ 1 & | z| \ge \sqrt{\delta \rho }\end{array}\right.} \) 
and for all \( \xi \in W^{1,p}(B_{\rho |\delta _k|})\)  satisfying \(\xi (0) = 0\)
\( \Vert (\nabla \beta _{2;\rho ,\delta }) \xi \Vert _{0,p} \le c\log ( \rho ^2 )^{-1 + 1/p} \Vert \xi \Vert _{1,p}, \quad \Vert \beta _{2;\rho ,\delta } \Vert _{1,2} \le C \log (\rho ^{2})^{-1/2}.\) 

Recall the map \({u}_0\)  from Remark REF .
Lemma 3.3.6 
For sufficiently small \({\delta }\)  there exists a right inverse
\(Q_{{u}_0^{{\delta }}}\)  of \(\tilde{D}_{{u}_0^{{\delta }}}\) 
with image the \(L^2\) -perpendicular of the kernel of
\(\tilde{D}_{{u}_0^{{\delta }}}\) .

Consider the maps defined by parallel transport using the modified
Levi-Civita connection,
\(\Pi _{{u}_0^{{\delta }}}^{{u}}: \Omega ^0({\Sigma },({u}_0^{{\delta }})^* TX) \rightarrow \Omega ^0({\Sigma },({u}^{{\delta }})^* TX) \) 
\( \Psi _{{u}_0^{{\delta }}}^{{u}}: \Omega ^{0,1}({\Sigma },({u}_0^{{\delta }})^* TX) \rightarrow \Omega ^{0,1}({\Sigma }, ({u}^{{\delta }})^* TX) .\) 
The operator \( \Psi _{{u}_0^{{\delta }}}^{{u}}\tilde{D}_{{\Sigma },{u}_0^{{\delta }}}(\Pi _{{u}_0^{{\delta }}}^{{u}})^{-1} \)  approaches the
operator \(\tilde{D}_{{u}}\)  as \({\delta } \rightarrow 0\) , c.f.
[2]}. The statement of the lemma
follows.
Define the approximate right inverse for
\(\tilde{D}_{{u}}^{{{\delta }}}\)  by composing the right inverse
\(Q_{{u}_0^{{\delta }}}\)  with a cutoff and extension operator:
\( T_{{\delta }} := P_{{\delta }} Q_{{u}_0^{{\delta }}}K_{{\delta }} ,\) 
defined as follows. The cutoff operator
\( K_{{\delta }}: \Omega ^{0,1}({u}^{{\delta }}* TX)_{0,p,{\delta }} \rightarrow \Omega ^{0,1}({u}_0^{{\delta },*} TX)_{0,p} \) 
is defined by
\( (K_{{\delta }}(\eta ))(z) ={\left\lbrace \begin{array}{ll} \eta (z) & z \notin \bigcup _{k,\pm } B_{| \delta _k(z)|^{1/2}} (w_k^\pm ) \\0 & \text{otherwise} \end{array}\right.} .\) 
We have
\( \Vert K_{{\delta }} \eta \Vert _{0,p}\le \Vert \eta \Vert _{0,p,{\delta }} \) 
by definition of the \(0,p,{\delta }\)  norm. The extension operator
\( P_{{\delta }}: \operatorname{Def}_\Gamma ({\Sigma }) \oplus \Omega ^0({\Sigma },{u}_0^{{\delta }}* TX)_{1,p,{\delta }} \rightarrow \operatorname{Def}({\Sigma }^{{\delta }}) \oplus \Omega ^0({\Sigma }^{{\delta }},{u}^{{\delta },*}TX)_{1,p,{\delta }} \) 
is defined as follows. For each component \(\Sigma _i\)  let \(\Sigma _i^*\) 
denote the complements of small balls around the nodes
\( \Sigma _i^* = \Sigma _i - \bigcup _{l, w_l^\pm \in \Sigma _i} B_{ | \delta _l(z)|^{1/2}/\rho } (w_l^\pm ) \) 
and the inclusion \(\pi _i : \Sigma _i^* \rightarrow {\Sigma }^{{\delta }} \) 
induces a map \( \pi _{i,*} : \Omega ^0( \Sigma _i^*,u_i^* TX) \rightarrow \Omega ^0({\Sigma }^{{\delta }}, {u}^{{\delta },*} TX )_{0,p} .\) 
Define
\( P_{{\delta }}(\zeta ,\xi ) =(\zeta ^{{\delta }},\xi ^{{\delta }}) \) 
where \( \zeta ^{{\delta }}\)  is the image of \(\zeta \)  under the gluing
map (REF ) and \(\xi ^{{\delta }}\)  is obtained by patching
together the sections \(\xi \) ; on the gluing region arising from gluing
the \(k\) -th node \(w_k\)  the section \(\xi ^{{\delta }}\)  is given by the
sum
\( \pi _{i^+(k),*}\beta _{\rho ,\delta _k} (\xi _{i^+(k)} -\xi (w_k)) \\ + \pi _{i^-(k),*} \beta _{\rho ,\delta _k} (\xi _{i^-(k)} -\xi (w_k)) + \xi (w_k).\) 
Fix a metric \(\Vert \cdot \Vert \)  on the finite-dimensional space
\(\operatorname{Def}_\Gamma ({\Sigma })\)  and define
\(\Vert (\zeta ,\xi ) \Vert _{1,p,{\delta }} = \left( \Vert \zeta \Vert ^p+ \Vert \xi \Vert _{1,p,{\delta }}^p \right)^{1/p} .\) 
Proposition 3.3.7 
Let \({u} : {\Sigma } \rightarrow X\)  be a stable map. There exist
constants \(c,C > 0 \)  such that if \( \Vert {\delta } \Vert < c \)  then
the approximate inverse \( T_{{\delta }}\)  of (REF )
satisfies
\( \Vert (\tilde{D}_{{u}}^{{\delta }} T_{{\delta }} - I) \eta \Vert _{0,p,{\delta }} \le {{2}}\Vert \eta \Vert _{0,p,{\delta }},\quad \Vert T_{{\delta }} \Vert < C .\) 

By construction \(T_{{\delta }}\)  is an exact right
inverse for \(\tilde{D}_{{u}}^{{\delta }}\)  away from gluing region. In the
gluing region the variation of complex structure on the curve vanishes
and \(D_{{u}}^{{\delta }} = D_{x_k}\) , the standard Cauchy-Riemann operator
with values in \(T_{x_k}X\) . So
\( \tilde{D}_{{u}}^{{\delta }} T_{{\delta }} \eta - \eta &=& \sum D_{x_k}\beta _{2;\rho ,\delta }(z) ( \xi _{i^\pm (k)}(z) - \xi _{i^\pm (k)}(w_k)) \\&=& \sum (D_{x_k} \beta _{2;\rho ,\delta }(z) ) \xi _{i^\pm (k)}(z)\) 
since \(K_{{\delta }} \eta = 0 \)  on \(B_{| \delta _k|^{1/2}}(0)\)  in the
components adjacent to the node. Since \(p > 2\) , the
\(0,p,{\delta }\) -norm of the right hand side is controlled by the
ordinary \(L^p\)  norm. By (REF ) we have
\( \Vert \tilde{D}_{{u}}^{{\delta }} T_{{\delta }} \eta - \eta \Vert _{0,p,{\delta }} \le \sum _k c| \log ( \rho )|^{2/p-2} \Vert \xi _{i^\pm (k)}- \xi (w_k) \Vert _{1,p}.\) 
The last factor is bounded by \( \Vert K_{{\delta }} \eta \Vert _{0,p}\) , by
the uniform bound on \(Q_{{\delta }}\) , and hence \(\Vert \eta \Vert _{0,p,{\delta }}\) , by the uniform bound on \(K_{{\delta }}\) .
Define a right inverse \(Q_{{\delta }}\)  to
\(\tilde{D}_{{u}}^{{\delta }}\)  by the formula
\( Q_{{\delta }} = T_{{\delta }} (\tilde{D}_{{u}}^{{\delta }} T_{{\delta }})^{-1} = \sum _{k \ge 0}T_{{\delta }} (\tilde{D}_{{u}}^{{\delta }} T_{{\delta }} - I )^k .\) 
The uniform bound on \(T_{{\delta }}\)  from Lemma REF  implies
a uniform bound on \(Q_{{\delta }}\) .
Step 3: Uniform quadratic estimate
Proposition 3.3.8 
Let \({u}: {\Sigma } \rightarrow X\)  be a stable map, \({\delta }\)  a
collection of gluing parameters, and \({u}^{{\delta }}:{\Sigma }^{{\delta }} \rightarrow X\)  the approximate solution defined
above. For every constant \(c > 0\)  there exist constants \(c_0,\delta _0> 0\)  such that if \( {u} \in \operatorname{Map}({\Sigma },X)_{1,p}, \xi \in \Omega ^0({\Sigma }^{{\delta }},{u}^{{\delta },*}TX)_{1,p}\)
\( \Vert {\operatorname{d}}{u} \Vert _{0,p} \le c_0, \ \ \ \Vert \xi \Vert _{L_\infty } \le c_0 \ \ \ \ \Vert \zeta \Vert \le c_0,\ \ \ \ \Vert {\delta } \Vert < \delta _0\) 
then
\( \Vert (D \mathcal {F}_{{u}}^{{\delta }}(\zeta ,\xi , \zeta _1,\xi _1) -\tilde{D}_{{u}}^{{{\delta }}})(\zeta _1,\xi _1)\Vert _{0,p,{\delta }} \le c \Vert \zeta , \xi \Vert _{1,p} ( \Vert \zeta _1, \xi _1 \Vert _{1,p} .\) 

Here \(D \mathcal {F}_{{u}}^{{\delta }}(\zeta ,\xi ,\zeta _1,\xi _1)\)  denotes the derivative evaluated at \(\zeta ,\xi \) ,
applied to \(\zeta _1,\xi _1\) . We use similar notation throughout the
discussion. The proof uses uniform estimates on Sobolev embedding:
Lemma 3.3.9  There exists
a constant \(c > 0\)  independent of \(\delta \)  such that the embedding
\(\Omega ^0({\Sigma }^{{\delta }}, {u}^{{\delta },*}T X)_{{1,p},{\delta }}\rightarrow \Omega ^0({\Sigma }^{{\delta }}, {u}^{{\delta },*}T X)_{0,\infty } \) 
has norm less than \(c\) .

One writes the Sobolev norms as a contribution from
each component of the curve \({\Sigma }\) . Then on each piece,
the metric near the boundary is uniformly comparable with
the flat metric. The claim then follows from
[7]} which shows that the constants
in the Sobolev embeddings depend only on the dimensions of the cone in
the cone condition.
For simplicity we assume a single gluing parameter \(\delta \) .
Let
\( \Psi _{{u}}^{{\delta },x}(\zeta ,\xi ): \Lambda ^{0,1} T_z^*{\Sigma }^{\delta } \otimes T_x X \rightarrow \Lambda ^{0,1}_{{j}^{{\delta }}(\zeta )} T_z^*{\Sigma } \otimes T_{\exp _x(\xi )} X \) 
denote pointwise parallel transport as in (REF ) using the
parallel transport using the modified Levi-Civita connection, and
projecting onto the \(0,1\) -part of the form defined using the complex
structure \({j}^{{\delta }}(\zeta )\)  obtained from gluing
\({j}(\zeta )\) , see (REF ). Let
\( \Theta _{{u}}^{{\delta },x}(\zeta ,\xi ,\zeta _1,\xi _1;\eta ) =\tilde{\nabla }_t \Psi _{{u}}^{{\delta },x} ( \zeta + t \zeta _1, \xi +t\xi _1) \eta .\) 
For \(\xi , \eta \)  sufficiently small there exists a constant \(c\)  such
that
\( | \Theta _{{u}}^{\delta ,x}(\zeta ,\xi ,\zeta _1,\xi _1; \eta )| \le c \Vert \xi ,\zeta \Vert \Vert \xi _1,\zeta _1 \Vert \Vert \eta \Vert \) 
where the norms on the right-hand side are any norms on the finite
dimensional vector spaces \(T_{{\Sigma }} M_{g,n,\Gamma }\)  and \(T_xX\) . This estimate is uniform in \(\delta \) , for the variation in
complex structure vanishes in a neighborhood of the nodes.
Differentiate the equation
\( \Psi _{{u}}^{\delta }(\zeta ,\xi ) \mathcal {F}_{{u}}^{\delta }(\zeta ,\xi )= \overline{\partial }_{{j}^\delta (\zeta )}(\exp _{{u}^\delta }(\xi ))) \) 
with respect to \((\zeta _1,\xi _1)\)  to obtain
\(\Theta _{{u}^\delta }(\zeta ,\xi ,\zeta _1,\xi _1,\mathcal {F}_{{u}}^{\delta }(\zeta ,\xi ) ) +\Psi _{{u}}^{\delta }(\zeta ,\xi )( D\mathcal {F}_{{u}}^{\delta }(\zeta ,\xi ,\zeta _1,\xi _1)) =\\ \tilde{D}_{{u}}^{\delta }(\xi , Dj^\delta (\zeta ,\zeta _1),D\exp _{{u}^\delta } (\xi ,\xi _1)) .\) 
Using the pointwise inequality
\( | \mathcal {F}_{{u}}^{\delta }(\zeta ,\xi ) | < c | {\operatorname{d}}{\exp _{{u}^\delta }(\xi )} | < c ( | {\operatorname{d}}{u}^\delta | + | \nabla \xi | )\) 
for \(\zeta ,\xi \)  sufficiently small, the estimate (REF ) on
\(\Phi \)  produces a pointwise estimate
\( | (\Psi _{{u}}^{\delta })^{-1}(\xi )\Theta _{{u}}^{\delta }(\zeta ,\xi ,\zeta _1,\xi _1,\mathcal {F}_{{u}}^{\delta }(\zeta ,\xi ))| \le c (| {\operatorname{d}}{u}^\delta | + | \nabla \xi |) \, | ( \xi ,\zeta ) |\, | (\xi _1,\zeta _1) | .\) 
Hence
\( \Vert \Psi _{{u}^\delta }^{-1}(\xi )\Theta _{{u}}^{\delta }(\zeta ,\xi ,\zeta _1,\xi _1,\mathcal {F}^\delta _{{u}}(\zeta ,\xi ))\Vert _{0,p} \\ \le c ( 1+ \Vert {\operatorname{d}}{u}^\delta \Vert _{0,p} + \Vert \nabla \xi \Vert _{0,p} ) \Vert (\xi ,\zeta )\Vert _{0,\infty } \Vert (\xi _1,\zeta _1) \Vert _{0,\infty }.\) 
It follows that
\( \Vert \Psi _{{u}}^{\delta }(\xi )^{-1}\Theta _{{u}}^{\delta }(\zeta ,\xi ,\zeta _1,\xi _1,\mathcal {F}_{{u}}^{\delta }(\zeta ,\xi )) \Vert _{0,p}\le c \Vert (\xi ,\zeta ) \Vert _{1,p} \Vert (\xi _1,\zeta _1) \Vert _{1,p}\) 
since the \(W^{1,p}\)  norm controls the \(L^\infty \)  norm by Lemma REF .
We next show that there exists a constant \(c > 0\)  such that uniformly
in \(\delta \) ,
\(  \Vert \Psi _{{u}^\delta }(\xi )^{-1}\tilde{D}_{{u}}^\delta (\xi , D{j}^{{\delta }}(\zeta ,\zeta _1),D\exp _{{{u}^\delta }} (\xi ,\xi _1)) - \tilde{D}_{{u}}^{\delta }(\zeta _1,\xi _1) \Vert _{0,p}\le c \Vert \zeta ,\xi \Vert _{1,p} \Vert \zeta _1,\xi _1 \Vert _{1,p}.\) 
Indeed differentiate (REF ) to obtain
\(\tilde{D}_{{u}}^{\delta }(\xi , D {j}^\delta (\zeta ,\zeta _1),D\exp _{{{u}^\delta }}(\xi ,\xi _1)) = \nabla ^{0,1}_{j^\delta (\zeta )}D\exp _{{{u}^\delta }}(\xi ,\xi _1) - \\ {{2}}\pi ^{0,1}_{j^\delta (\zeta )}J_{\exp _{{u}^\delta }(\xi )} {\operatorname{d}}\exp _{{u}^\delta }(\xi )D{j}^\delta (\zeta ,\zeta _1)  - J_{\exp _{{u}^\delta }(\xi )} (\nabla _{D\exp _{{{u}}}(\xi ,\xi _1)} J_{\exp _{{u}^\delta }(\xi )} )\partial \exp _{{u}^\delta }(\xi ) . \) 
Hence
\( \tilde{D}_{{u}}^\delta (\xi , D{j}^\delta (\zeta ,\zeta _1),D\exp _{{{u}^\delta }}(\xi ,\xi _1))- \Psi _{{u}^\delta }(\xi ) \tilde{D}_{{u}}^{\delta }(\zeta _1,\xi _1)= \Pi _1 + \Pi _2 + \Pi _3 \) 
where the three terms \(\Pi _1,\Pi _2,\Pi _3\)  are
\( \Pi _1 =\nabla ^{0,1}_{{j}^\delta (\zeta )} D\exp _{{{u}^\delta }}(\xi ,\xi _1)-\Psi _{{u}^\delta }(\xi ) \nabla ^{0,1}_{{j}^\delta (0)} \xi _1\) 
\( \Pi _2 = - {{2}}\pi ^{0,1}_{{j}^\delta (\zeta )} J_{\exp _{{u}}(\xi )} {\operatorname{d}}\exp _{{u}^\delta }(\xi ) D{j}^\delta (\zeta ,\zeta _1) +\Psi _{{u}^\delta }(\xi ) \pi ^{0,1}_{{j}^\delta (0)} {{2}}J_{{u}^\delta } {\operatorname{d}}{u}^\delta D{j}^\delta (0,\zeta _1)\) 
\( \Pi _3 = - {{2}}J_{\exp _{{u}}(\xi )} (\nabla ^{0,1}_{D\exp _{{{u}}^\delta }(\xi ,\xi _1)} J_{\exp _{{u}^\delta }(\xi )} ) \partial _{{j}^\delta (\zeta )}\exp _{{u}^\delta }(\xi ) + {{2}}\Psi _{{u}^\delta }(\xi ) J_{{u}^\delta }(\nabla _{\xi _1} J_{{u}} ) \partial _{{j}^\delta (0)} {u}^\delta .\) 
The first difference has norm given by
\( | \pi _{{j}^\delta (\zeta )}^{0,1} ( \nabla ( D \exp _{{{u}^\delta }}(\xi ,\xi _1))- \Psi _{{u}^\delta }(\xi ) \nabla \xi _1 ) | \\ \le |\pi _{{j}^\delta (\zeta )}^{0,1} ( \nabla ( D \exp _{{{u}^\delta }}(\xi ,\xi _1)) -D\exp _{{{u}^\delta }}(\xi ,\nabla \xi _1) )| + |\pi ^{0,1}_{{j}^\delta (\zeta )} (D\exp _{{{u}^\delta }}(\xi ,\nabla \xi _1) - \Psi _{{u}^\delta }(\xi )\nabla \xi _1 ) | \\ \le c | \nabla \xi | | \xi _1| + c ( | \zeta | + |\xi |) |\nabla \xi _1| + c | {\operatorname{d}}{u}^\delta | | \xi | | \xi _1|.\) 
We write for the second difference
\(| \pi ^{0,1}_{{j}^\delta (\zeta )}( J_{\exp _{{u}^\delta }(\xi )} {\operatorname{d}}\exp _{{u}^\delta }(\xi ) D {j}^\delta (\zeta ,\zeta _1) -\Psi _{{u}^\delta }(\xi ) J_{{u}^\delta } {\operatorname{d}}{u}^\delta D{j}^\delta (0,\zeta _1))| \\\le | \pi ^{0,1}_{{j}^\delta (\zeta )} ( J_{\exp _{{u}^\delta }(\xi )} {\operatorname{d}}\exp _{{u}^\delta }(\xi ) D{j}^\delta (\zeta ,\zeta _1) -J_{\exp _{{u}^\delta }(\xi )} \Psi _{{u}^\delta }(\xi ) {\operatorname{d}}{u}^\delta D{j}^\delta (0,\zeta _1))|+ \\|\pi ^{0,1}_{{j}^\delta (\zeta )} ( J_{\exp _{{u}^\delta }(\xi )} \Psi _{{u}^\delta }(\xi ) {\operatorname{d}}{u}^\delta D{j}^\delta (0,\zeta _1) -\Psi _{{u}^\delta }(\xi ) J_{{u}} {\operatorname{d}}{u}^\delta D{j}^\delta (0,\zeta _1))| \\\le c ( |\zeta | + |\xi | + | {\operatorname{d}}{u}^\delta | + | \nabla \xi |) | |\zeta _1| .\) 
The third term can be estimated pointwise by
\( | J_{\exp _{{u}^\delta }(\xi )}(\nabla _{D \exp _{{u}^\delta }(\xi ,\xi _1)}J_{\exp _{{u}^\delta }(\xi )} ) \partial _{{j}^\delta (\zeta )} \exp _{{u}^\delta }(\xi ) -\Psi _{{u}^\delta }(\xi ) J_{{u}^\delta }(\nabla _{\xi _1} J_{{u}^\delta } ) \partial _{{j}^\delta (0)}{u}^\delta | \) 
\( \le | J_{\exp _{{u}^\delta }(\xi )}(\nabla _{D \exp _{{{u}}}(\xi ,\xi _1)} J_{\exp _{{u}^\delta }(\xi )} )\partial _{{j}^\delta (\zeta )} \exp _{{u}^\delta }(\xi ) -J_{\exp _{{u}^\delta }(\xi )} (\nabla _{D\exp _{{u}^\delta }(\xi ,\xi _1)} J_{\exp _{{u}^\delta }(\xi )} )\partial _{{j}^\delta (0)} \exp _{{u}^\delta }(\xi )| \) 
\( +| J_{\exp _{{u}^\delta }(\xi )}(\nabla _{D \exp _{{{u}}}(\xi ,\xi _1)} J_{\exp _{{u}^\delta }(\xi )}) \partial _{{j}^\delta (0)} \exp _{{u}^\delta }(\xi )-\Psi _{{u}^\delta }(\xi ) J_{{u}^\delta }(\nabla _{\xi _1} J_{{u}^\delta }) \partial _{{j}^\delta (0)} {u}^\delta | \) 
\( \le c | \zeta | (| {\operatorname{d}}{u}^\delta | + |\nabla \xi |) |\xi _1| + c ( |{\operatorname{d}}{u}^\delta | + | \nabla \xi |) ( | \xi _1 |) \) 
for \(\xi \)  sufficiently small. Combining these estimates and
integrating, using the \(0,p,\delta \) -norms on \({\operatorname{d}}{u}\) , \(\nabla \xi ,\nabla \xi _1\)  and the \(L^\infty \)  norms on the other factors and Lemma
REF , completes the proof.
Step 4: Implicit Function Theorem
For any \((\zeta _0,\xi _0) \in \operatorname{Def}_\Gamma ({u})\)  we denote by
\(\zeta _0^{{\delta }}\)  the deformation of \({\Sigma }^{{\delta }}\) 
defined in (REF ) and by \(\xi _0^{{\delta }}\)  the section of
\({u}^{{\delta },*} TX\)  defined as in (REF ).
Theorem 3.3.10 
Let \({u}:{\Sigma } \rightarrow X\)  be a stable map.
There exist constants
\(\epsilon _0,\epsilon _1 > 0\)  such that for any \((\zeta _0,\xi _0,{\delta }) \in  \operatorname{ker}\tilde{D}_{{u}} \times \mathbb {R}^m\)  of norm at most \(\epsilon _0\) , there is a
unique
\((\zeta _1,\xi _1) = (\tilde{D}_{{u}}^{{\delta }})^* \eta _1 \) 
of norm at most \(\epsilon _1\)  such that the map
\(\exp _{{u}^{{\delta }}}(\xi _0^{{\delta }} + \xi )\)  is
\({j}^\delta (\zeta _0 + \zeta _1)\) -holomorphic, and depends smoothly on
\(\zeta _0,\xi _0\) .

The first claim is an application of the quantitative version of the implicit
function theorem (see for example [2]}) using
the uniform error bound from Proposition REF , uniformly
bounded right inverse from Proposition REF , and uniform
quadratic estimate from Proposition REF .
Step 5: Rigidification
In the previous step we have constructed a family of stable maps which
we will show eventually gives rise to a parametrization of all nearby
stable maps. A more natural way of parametrizing nearby stable maps
involves examining the intersections with a family of codimension two
submanifolds. For example, this construction of charts is that given
in the algebraic geometry approach of Fulton-Pandharipande
[14]}. In order to carry this out in the symplectic approach,
we study the differentiability of the evaluation maps. Let \(u_S:\Sigma _S \rightarrow X\)  over a parameter space \(S \subset \operatorname{Def}({u})\)  be
the family of maps defined in the previous step. The following is
similar to [15]}.
Theorem 3.3.11  If \( {u}_S\) 
is constructed using the exponential gluing profile \(\varphi \)  and \(U\subset {\Sigma }\)  is an open neighborhood of the nodes then the
map \((z,s) \mapsto {u}_s(z)\)  is differentiable on a neighborhood
of \(({\Sigma } - U ) \times \lbrace  0 \rbrace \) .

For simplicity, we assume that there is a single gluing parameter
\(\delta \) . Differentiability for \(\delta \)  is studied in
McDuff-Salamon [2]}. The discussion in our
case is somewhat easier, because we use a fixed right inverse in the
gluing construction. Given \((\zeta ,\xi ) \in \operatorname{Def}_\Gamma ({\Sigma })\times \Omega ^0( {\Sigma }^\delta , ({u}^\delta )^* TX)\) , we
constructed a unique correction \((\zeta _1,\xi _1)\)  in the image of the
right inverse such that
\( \overline{\partial }_{{j}^\delta (\zeta _0 + \zeta _1)} \exp _{{u}^\delta }(\xi _0 +\xi _1) = 0 .\) 
For \(\delta \)  fixed, \((\zeta _1,\xi _1)\)  depends smoothly on
\((\zeta _0,\xi _0)\) , by the implicit function theorem. Hence the
evaluation at \(z \in {\Sigma } - U\)  also depends smoothly on
\((\zeta _0,\xi _0)\) .
The computation of the derivative with respect to the gluing parameter
is complicated by the fact that for each \(\delta \)  a different implicit
function theorem is applied to obtain the correction. Let
\(\tilde{D}_\delta = D \mathcal {F}_{{u}^{\delta }}\) . Differentiating the
equation \(\mathcal {F}_{{u}^{\delta }}(\zeta _0^\delta + \zeta _1,\xi _0^\delta + \xi _1) = 0 \)  with respect to \(\delta \)  gives
\( \tilde{D}_\delta \left( \frac{d}{d \delta }\zeta _1, D\exp _{{u}^\delta } (\xi _0^\delta +\xi _1 , 0 , \frac{d}{d \delta }\xi _1 ) \right) = - \tilde{D}_\delta \left( 0 ,D\exp _{{u}^\delta } (\xi _0^\delta + \xi _1 , \frac{d}{d \delta }{u}^\delta , 0 )\right) .\) 
From (REF ) we
have in the gluing region,
\(\frac{d}{d \delta }\overline{\partial }{u}^\delta &=& \frac{d}{d \delta }\overline{\partial }\exp _{x} ( \left(\alpha ( \rho \varphi ^{-1/2} |z|) \xi (z)) \right) \\&=&D\exp _{x} \left(\alpha ( \rho \varphi ^{-1/2} |z| ) \xi (z),\alpha ^{\prime }( \rho \varphi ^{-1/2} |z| )|z| \xi (z)\frac{d}{d \delta }\varphi ^{-1/2} \rho \right)^{0,1} .\) 
Hence there exists a constant \(C\)  depending on \(\rho ,\alpha \)  but not
on \(\delta \)  such that
\( \left| \frac{d}{d \delta }\overline{\partial }{u}^\delta \right| \le C \left|\frac{d}{d \delta }\varphi ^{-1/2} \right| .\) 
Now \({\operatorname{d}}\varphi \)  is given by
\( {\operatorname{d}}( e^{1/\delta } - e)^{-1/2} = (1/2) (e^{1/\delta } - e)^{-3/2} e^{1/\delta }\delta ^{-2} {\operatorname{d}}\delta = (1/2) ( e^{1/3\delta } - e^{-2/3\delta + 1})^{-3/2} \delta ^{-2} {\operatorname{d}}\delta .\) 
For \(\delta \)  small, this is less than \({{2}}e^{-1/2\delta } \delta ^{-2}\) . Integrating and
using the pointwise estimate (REF ) we obtain for some
constant \(C > 0\) ,
\( \left\Vert \frac{d}{d \delta }\overline{\partial }{u}^\delta _s \right\Vert _{0,p} \le Ce^{-1/2\delta } \delta ^{-2} \le Ce ^{-1 / \delta }\) 
for sufficiently small \(\delta \) . Now the uniform quadratic estimates
imply that \(\tilde{D}_\delta = D \mathcal {F}_{{u}^\delta }(\zeta ,\xi )\)  is
uniformly bounded from below on the right inverse of
\(\tilde{D}_{{u}}^{\delta } = D \mathcal {F}_{{u}}^{\delta }(0,0)\) , for
\((\zeta _0,\xi _0)\)  sufficiently small. It follows that
\( \left\Vert ( \frac{d}{d \delta }\zeta _1, \frac{d}{d \delta }\xi _1) \right\Vert _{1,p} \le Ce^{-1/\delta } \) 
for \(\zeta _0,\xi _0,\delta \)  sufficiently small as well. Hence the same is
true for the evaluation \(\frac{d}{d \delta }\xi _1(z)\)  for \(z \in {\Sigma } - U\) .
In particular,
\(\lim _{\delta \rightarrow 0 } (\partial _\delta \exp _{{u}^\delta }(\xi _0^\delta + \xi _1))(z) = 0. \) 
It follows that the differential of the evaluation map has a
continuous limit as \({\delta } \rightarrow 0\) , which completes the proof of the
Theorem.
Using the evaluation maps in the previous step, we construct
embeddings of the families constructed above into suitable moduli
spaces of stable marked curves, given by adding additional marked
points which map to fixed submanifolds in \(X\) . A codimension two
submanifold \(Y \subset X\)  is transverse to \({u}: {\Sigma }\rightarrow X\)  if \({u}\)  meets \(Y\)  transversally in a single point
\({u}(z)\) .
Definition 3.3.12 Let \({u}: {\Sigma } \rightarrow X\)  be a stable map.
Given any family \({Y} = (Y_1,\ldots , Y_\ell )\)  of codimension two
submanifolds transverse to \({u}\)  and a family
\({\Sigma }_S,{u}_S,{z}_S\)  with parameter space \(S\)  of an
\(n\) -marked stable map \(({\Sigma },{u},{z})\) , the rigidified family of \(n+\ell \) -marked nodal surfaces is defined by
\( {\Sigma }_S^{{Y},{u}} := ({\Sigma }_S,(z_{1,S}, \ldots , z_{n+\ell ,S})) \rightarrow S,\ \ \ \ {u}_s(z_{n+i,s}) \in Y_i .\) 

Proposition 3.3.13  Let \({u}_S\)  be a family of stable
maps over a parameter space \(S \subset \operatorname{Def}({u})\)  given by the
gluing construction using a gluing profile \(\varphi \)  and system of
coordinates \({\kappa }\) . Suppose that the evaluation map \(\operatorname{ev}:({\Sigma } - U) \times S \rightarrow X\)  is \(C^1\) , and that the rigidified
family has stable underlying curves. Then the rigidified family of
curves \({\Sigma }_S^{{Y},{u}}\)  is \(C^1\)  with respect to the
gluing profile and local coordinates, that is, the map
\( S \mapsto \overline{M}_{g,n+l}, \quad s \mapsto {\Sigma }_s^{{Y},{u}} \) 
is \(C^1\)  with respect to the smooth structure defined by \(\varphi ,{\kappa }\) .

By the implicit function theorem for \(C^1\)  maps
and differentiability of evaluation maps from the previous subsection.
Definition 3.3.14 
Let \({Y},{u}\)  be as above. The pair \(({Y},{u})\)  is compatible if

each \(Y_j\)  intersects \({u}\)  transversally in a single point
\(z_j \in {\Sigma }\) ;

if \(\xi \in  \operatorname{ker}(\tilde{D}_{{u}})\)  satisfies \(\xi (z_{n+j}) \in T_{{u}(z_{n+j})} Y_j\)  for \(j =1,\ldots , l\)  then \(\xi = 0\) ;

the curve \({\Sigma }\)  marked with the additional points
\(z_{n+1},\ldots , z_{n+\ell }\)  is stable;

if some automorphism of \(({\Sigma },{u})\)  maps
\(z_i\)  to \(z_j\)  then \(Y_i\)  is equal to \(Y_j\) .


The second condition says that there are no infinitesimal deformations
which do not change the positions of the extra markings.
Proposition 3.3.15 
Let \({u}\)  be a parametrized regular stable map, and \({u}_S\)  the
stratified-smooth universal deformation constructed in Theorem
REF  with base \(S \subset \operatorname{Def}({u})\) . There exists a
collection \({Y}\)  compatible with \({u}\) . Furthermore if the
evaluation map is \(C^1\)  as in Proposition REF  then \({\Sigma }^{{Y},{u}}_S\)  defines an \(C^1\) -immersion of \(S\)  into
\(\operatorname{Def}({\Sigma }^{{Y},{u}})\) .

First we show the existence of a compatible collection. Given a
regular stable map \(({\Sigma },{z} =(z_1,\ldots ,z_n),{u}:{\Sigma } \rightarrow X)\) , choose \(Y_1,\ldots ,Y_k\) 
transverse \({u}\)  on the unstable components of \({\Sigma }\) , so
that \({\Sigma }_1 = ({\Sigma }, (z_1,\ldots ,z_{n+k}))\)  is a
stable curve. Let \({\Sigma }_{S_1,1} \rightarrow S_1\)  denote a universal
deformation of \({\Sigma }_1\) . By universality, the family
\({\Sigma }_S^{{Y},{u}}\)  is induced by a map \(\psi : S \rightarrow S_1\) . We successively add marked points until \(\psi \)  is an immersion:
Suppose that \(\psi \)  is not an immersion. Then we may choose an
additional marked point \(z_{n+k+1} \in {\Sigma }\)  such that \({\operatorname{d}}\operatorname{ev}_{n+k+1}\)  is non-trivial on \( \operatorname{ker}D \psi \) . Since \({u}\)  is
holomorphic, \({\operatorname{d}}{u}(z_{n+k+1})\)  is rank two at \(z_{n+k+1}\) . Let
\(Y_{n+k+1} \subset X\)  be a codimension two submanifold containing
\({u}(z_{n+k+1})\)  such that \({u}\)  is transverse to \(Y_{n+k+1}\) 
at \(z_{n+k+1}\) , and \(Y_{n+k+1}\)  is transversal to \(\operatorname{ev}_{n+k+1} \)  at
\({\Sigma },{u}\) . Suppose \(z_{n+n^{\prime }+1}\)  has orbit \(z_{n+k+1},z_{n+k+2},\ldots , z_{n+l}\)  under the group \( \operatorname{Aut} ({u})\) . Repeating
the same submanifold for each marking related by automorphisms gives
a collection invariant under the action of automorphisms. The map
\(\psi _1\)  for the new family has property that the dimension of
\( \operatorname{ker}(D\psi _1)\)  has dimension at least two less than that of
\( \operatorname{ker}(D\psi )\) . It follows that the procedure terminates after adding
a finite number of markings. The last claim follows from the second
condition in Definition REF .
Step 6: Surjectivity
In this section, we show that the family constructed above contains a
Gromov neighborhood of the central fiber. First we show:
Proposition 3.3.16  There exists a constant \(\epsilon > 0\) 
such that any stable map \(({\Sigma }_1,{u}_1)\)  with complex
structure on \({\Sigma }_1\)  given by \(j^\delta (\zeta )\)  for some
\(\zeta \in \operatorname{Def}({\Sigma }^{{\delta }})\) , and \( {u}_1 :=\exp _{{u}^{{\delta }}}({\xi }) \)  with
\( \Vert \zeta \Vert ^2 + \Vert \xi \Vert ^2_{1,p,{\delta }} < \epsilon \) 
is of the form in Theorem REF  for some \((\xi _1,\zeta _1)\in \operatorname{Im}(\tilde{D}_{{u}}^{{{\delta }}})^*\) .

Compare with [2]}.
Let \((\zeta ,{\xi })\)  be as in the statement of the Proposition. We
claim that \((\zeta ,{\xi }) =(\zeta _0^{{\delta }},{\xi }_0^{{\delta }}) +(\zeta _1,{\xi }_1)\)  for some \((\zeta _0,{\xi }_0) \in  \operatorname{ker}(\tilde{D}_{{u}})\)  and \((\zeta _1,{\xi }_1) \in \operatorname{Im}((D_{{u}}^{{{\delta }}})^*)\)  with small norm. It then follows
by the implicit function theorem that \((\zeta _1,{\xi }_1)\)  is the
solution given in Theorem REF . Now \(\zeta =\zeta _0^{{\delta }}\)  for some \(\zeta _0 \in \operatorname{Def}_\Gamma ({\Sigma })\)  and gluing parameters \({\delta }\) ,
because \(\operatorname{Def}({\Sigma }^{{\delta }})\)  is the direct sum of the
image of \(\operatorname{Def}_\Gamma ({\Sigma })\)  and \(m\) . By the gluing
theorem for indices (see e.g. [18]}), the
image of \(\operatorname{Def}_\Gamma ({u})\)  under the gluing map projects
isomorphically onto \( \operatorname{ker}(\tilde{D}_{{u}}^{{\delta }})\)  for
\({\delta }\)  sufficiently small, and so \(\operatorname{Def}_\Gamma ({u})\)  is
transverse to \(\operatorname{Im}(\tilde{D}_{{u}}^{{\delta }})^*\) , for
\({\delta }\)  sufficiently small. The claim then follows from the
inverse function theorem.
Given a regular stable \({u}\)  with stable domain, consider the
family of \(J\) -holomorphic maps \( {u}_S\)  produced by Theorem
REF  with parameter space a neighborhood \(S\)  of 0 in
\(\operatorname{Def}({u}),\)  equipped with a canonical identification \(\iota \)  of
the central fiber with the original map \({u}\) . In the case that
the domain \({\Sigma }\)  is not a stable (marked) curve, we choose
codimension two submanifolds \({Y} = (Y_1,\ldots , Y_l)\)  meeting
\({u}\)  transversally so that \({\Sigma }\)  with the additional
marked points is stable. Applying this to the family \({u}_S\)  gives
a family of marked stable maps \({u}_S^{{Y}}\)  with \(n + l\)  marked
points over a parameter space \(S \subset \operatorname{Def}({u}^{{Y}})\)  in the
deformation space of the map with the additional marked points. Now
\(\operatorname{Def}({u}^{{Y}}) \cong \operatorname{Def}({u}) \oplus \bigoplus _{i=1}^lT_{z_i} {\Sigma }\)  includes the deformations of the markings, but
these are fixed by requiring that the additional marked points map to
the given collection \({Y}\) . Forgetting the additional marked
points gives a family \({u}_S\)  of stable maps with \(n\)  marked points
over a neighborhood of 0 in \(\operatorname{Def}({u})\) .
Proposition 3.3.17 \(({u}_S,\iota )\)  is a versal stratified-smooth
deformation of \({u}\) , and in fact \({u}_S\)  gives a versal
stratified-smooth deformation of any of its fibers.

First suppose that \({\Sigma }\)  is stable.
Let \(({u}_{S^1}^1,\iota ^1)\)  be another stratified-smooth
deformation of \({u}\)  with parameter space \(S^1\) . Let
\({\Sigma }_S \rightarrow S \subset \operatorname{Def}({\Sigma })\)  be a minimal versal
deformation of \({\Sigma }\) . The family \({\Sigma }_{S^1}^1\)  is
obtained by pull-back of \({\Sigma }_S\)  by a stratified-smooth map
\(\psi : S^1 \rightarrow S\) . By definition the map \({u}_s^1\)  converges to
the central fiber in the Gromov topology as \(s\)  converges to the base
point \(0 \in S^1\) . The exponential decay estimate of [2]} for holomorphic cylinders of small energy imply that
for \(s\)  sufficiently close to 0, \({\Sigma }_s^1,{u}_s^1\)  is
given by exponentiation, \( {u}_s^1 =\exp _{{u}^{{\delta }}}({\xi }) \)  for some \({\xi } \in \Omega ^0( {u}^{{\delta },*} TX)\)  with \(\Vert {\xi }\Vert _{1,p} < \epsilon _1\) , for \(s\)  sufficiently close to 0. Proposition
REF  produces a stratified-smooth map \(\psi : S^1 \rightarrow \operatorname{Def}({u})\)  such that \({u}_{S^1}^1\)  is the pull-back of \(\psi \) .
To show that the deformation \(({u}_S, \iota )\)  is universal, let \(\phi _j: {\Sigma }_{S^1}^1 \rightarrow \psi _j^* {\Sigma }_S, j = 0,1 \)  be
isomorphisms of families inducing the identity on the central fiber.
The difference between the two automorphisms is an automorphism of the
family \({\Sigma }_{S^1}^1\)  inducing the identity on the central
fiber; since the automorphism group of the central fiber is discrete,
the automorphism must be the identity. In the case that
\({\Sigma }\)  is not stable, after adding marked points passing
through \(Y_1,\ldots , Y_l\) , we obtain a family
\({u}_{S^1}^{1,{Y}}\)  of stable maps with \(n+l\)  marked points.
By the case with stable domain, this family is obtained by pull-back
of \({u}_S^{{Y}}\)  by some map \(S^1 \rightarrow S\) . Hence
\({u}_{S^1}^1\)  is obtained by pull-back by the same map. The
argument for an arbitrary fiber is similar and left to the reader.
Remark 3.3.18 In the case that \({\Sigma }\)  is unstable, it seems
likely that restricting the family of Theorem REF  to
\(\operatorname{Def}({u})\)  (that is, the perpendicular of \( \operatorname{aut} ({\Sigma })\) )
also gives a universal deformation, but we do not know how to prove
this. The problem is that in this case, several different gluing
parameters give the same curve, and we do not have an implicit
function theorem for varying gluing parameter.

Step 7: Injectivity
By injectivity, we mean that the family constructed above contains
each nearby stable map exactly once, up to the action of \( \operatorname{Aut} (u)\) .
This is part of what we called “strongly universal” in Definition
REF .
Theorem 3.3.19 The versal deformations constructed in Step 6 above
are strongly universal.

Let \({u}_S\)  be a deformation constructed
as in Step 6, using the exponential gluing profile. Let
\({\Sigma }_{1,S^1} \rightarrow S^1\)  be a family giving a universal
deformation of the curve \({\Sigma }^{{Y},{u}}\)  obtained by
adding the additional markings mapping to the given submanifolds. By
Definition REF , the family \({\Sigma }_{S}^{{Y},{u}}\) 
induces a map \(\phi : S \rightarrow S_1\)  whose differential is injective in a
neighborhood of 0. By the inverse function theorem for \(C^1\)  maps,
\(\phi \)  induces a homeomorphism onto its image. In particular, any two
distinct fibers of \({\Sigma }_S^{{Y},{u}}\)  are non-isomorphic,
and so two fibers of \({\Sigma }_S\)  are isomorphic if and only if
they are related by a permutation of the markings. After shrinking
\(S\) , this happens only if the permutation is induced by an
automorphism of \({u}\) . Given another family \({u}_{S^{\prime }}^{\prime } :{\Sigma }_{S^{\prime }} \rightarrow S^{\prime }\)  corresponding to a deformation of a fiber of
\({u}_S \rightarrow S\) , by the uniqueness part of the implicit function
theorem, a map \(\phi ^{\prime }: S^{\prime } \rightarrow S_1\)  so that \({u}_{S^{\prime }}^{\prime }\)  is obtained
by pull-back from \({u}_S\) , and this map is unique by the
injectivity just proved. This shows that \({u}_S\)  gives a
stratified-smooth universal deformation of any of its fibers, and so
is strongly universal.
The Theorem implies that the families in the universal deformations
constructed above define stratified-smooth-compatible charts for the
moduli space \(\overline{M}_{g,n}(X,d)\) . That is, for any stratum
\(M_{g,n,\Gamma }(X,d)\) , the restriction of the charts given by the
universal deformation of some map of type \(\Gamma \)  to
\(M_{g,n,\Gamma }(X,d)\)  are smoothly compatible.
Corollary 3.3.20 
Let \(X,J\)  be as above. For any \(g \ge 0, n \ge 0\) , the strongly
universal stratified-smooth deformations of parametrized regular
stable maps provide \(\overline{M}_{g,n}^{{\operatorname{reg}}}(X)\)  with the structure of a
stratified-smooth topological orbifold.

In order to apply localization one needs to know that the fixed point
sets admit tubular neighborhoods. For this it is helpful to know that
\(\overline{M}_{g,n}^{\operatorname{reg}}(X,d)\)  admits a \(C^1\)  structure. In order to obtain
compatible charts, we construct the local coordinates inductively as
in Definition REF , starting with the strata of highest
codimension.
Proposition 3.3.21 
Let \(X,J\)  be as above. For any compatible system of local coordinates
near the nodes, the strongly universal deformations constructed using
the exponential gluing profile equip \(\overline{M}_{g,n}^{\operatorname{reg}}(X)\)  with the
structure of a \(C^1\) -orbifold.

We claim that the charts induced by the universal
deformations are \(C^1\) -compatible, assuming they are constructed from
the same system of local coordinates near the nodes. Given two sets
of submanifolds \({Y}_1,{Y}_2\) , define \({Y} = {Y}_1 \cup {Y}_2\) . The family \( {\Sigma }^{{Y},{u}}\)  admits proper
étale forgetful maps \( {\Sigma }^{{Y},{u}}_S \rightarrow {\Sigma }_S^{{Y}_j,{u}} , \quad j= 1,2 .\)  The fiber consists
of reorderings of the additional marked points induced by the action
of \( \operatorname{Aut} ({\Sigma },{u})\) , and the diagram provided by
\({\Sigma }^{{Y},{u}}\)  expresses the composition as a smooth
\(C^1\) -morphism of orbifolds.
Remark 3.3.22  Any compact \(C^1\)  orbifold admits a compatible
\(C^\infty \)  structure, in analogy with the situation with manifolds.
Indeed, as is well known any orbifold admits a presentation as the
quotient of a manifold (namely its orthogonal frame bundle) by a
locally free group action, and so the orbifold case follows from the
equivariant case proved in Palais [20]}. Hence
\(\overline{M}^{{\operatorname{reg}}}_{g,n}(X,d)\)  if compact admits a (non-canonical)
smooth structure. Presumably the compactness assumption may be
removed but we have not proved that this is so. See however the
construction of smoothly compatible Kuranishi charts in
[15]}.


Deformations of symplectic vortices
We begin by reviewing the theory of symplectic vortices introduced by
Mundet i Riera [22]} and Salamon and collaborators
[23]}. Let \(\Sigma \)  be a compact complex curve, \(G\)  a
compact Lie group, and \(\pi : P \rightarrow \Sigma \)  a smooth principal
\(G\) -bundle. Given any left \(G\) -manifold \(F\)  we have a left action of
\(G\)  on \(P \times F\)  given by \(g(p,f) = (pg^{-1},gf)\)  and we denote by
\(P(F) = (P \times F)/G\)  the quotient, that is, the associated fiber
bundle with fiber \(F\) . Let \(X\)  be a compact Hamiltonian \(G\) -manifold
with symplectic form \(\omega \)  and moment map \(\Phi :X \rightarrow \mathfrak {g}^*\) . The
action of \(G\)  on \(X\)  induces an action on \(J(X)\) ; and we denote by
\(J(X)^G\)  the invariant subspace. Let \(\psi : \Sigma \rightarrow BG\)  be a
classifying map for \(P\) , so that \(P \cong \psi ^* EG\)  and \(P(X) \cong \psi ^* EG \times _G X \cong \psi ^* X_G\)  where \(X_G = EG \times _G X\) .
Continuous sections \(u: \Sigma \rightarrow P(X)\)  are in one-to-one
correspondence with lifts of \(\psi \)  to \(X_G\) . The homology class
\(\deg (u)\)  of the section \(u\)  is defined to be the homology class
\(\deg (u) \in H_2^G(X,\mathbb {Z})\)  of the corresponding lift. Let \(\mathcal {A}(P)\)  be
the space of smooth connections on \(P\) , and \(P(\mathfrak {g})\)  the adjoint
bundle. For any \(A \in \mathcal {A}(P)\) , let \(F_A \in \Omega ^2(\Sigma ,P(\mathfrak {g}))\) 
the curvature of \(A\) . Any connection \(A \in \mathcal {A}(P)\)  induces a map of
spaces of almost complex structures
\( J(X)^G \rightarrow J(P(X)), \ \ J \mapsto J_A\) 
by combining the almost complex structures on \(X\)  and \(\Sigma \)  using
the splitting defined by the connection. Let \(\Gamma (\Sigma ,P(X))\) 
denote the space of smooth sections of \(P(X)\) . Consider the vector
bundle
\( \bigcup _{u \in \Gamma (\Sigma ,P(X)) } \Omega ^{0,1}(\Sigma ,u^*T^{\operatorname{vert}} P(X)) \rightarrow \Gamma (\Sigma ,P(X)) .\) 
We denote by \(\overline{\partial }_A\)  the section given by the Cauchy-Riemann operator
defined by \(J_A\) . A gauged map from \(\Sigma \)  to \(X\)  is a datum
\((P,A,u)\)  where \(A \in \mathcal {A}(P)\)  and \(u: \Sigma \rightarrow P(X)\)  is a section.
A gauged holomorphic map is a gauged map \((P,A,u)\)  such that
\(\overline{\partial }_A u = 0 \) . Let \(\mathcal {H}(P,X)\)  be the space of gauged holomorphic
maps with underlying bundle \(P\) . Let \(\mathcal {G}(P)\)  denote the group of
gauge transformations
\( \mathcal {G}(P) = \lbrace  a: P \rightarrow P, a(pg) = a(p)g, \ \ \ \pi \circ a = \pi \rbrace  .\) 
The Lie algebra of \(\mathcal {G}(P)\)  is the space of sections
\(\Omega ^0(\Sigma ,P(\mathfrak {g}))\)  of the adjoint bundle \(P(\mathfrak {g}) = P \times _G\mathfrak {g}\) . We identify \(\mathfrak {g}\rightarrow \mathfrak {g}^*\) , and hence \(P(\mathfrak {g}) \rightarrow P(\mathfrak {g}^*)\) , using
an invariant metric on \(\mathfrak {g}\) . Let \(P(\Phi ): P(X) \rightarrow P(\mathfrak {g})\)  denote the
map induced by the equivariant map \(\Phi : X \rightarrow \mathfrak {g}\) .
Definition 4.0.23 A gauged holomorphic map \((A,u) \in \mathcal {H}(P,X)\)  is a symplectic
vortex (or vortex for short) if it satisfies
\( F_A +  \operatorname{Vol}_{\Sigma } u^* P(\Phi ) = 0 .\) 
An \(n\) -marked symplectic vortex is a vortex \((A,u)\)  together
with \(n\) -tuple \({z} = (z_1,\ldots , z_n)\)  of distinct points on
\(\Sigma \) . A marked vortex \((A,u,{z})\)  is stable if it has
finite automorphism group.

The equation in the definition can be interpreted as the zero level
set condition for a formal moment map for the action of the group of
gauge transformations, see [22]}, [23]}. The energy of a gauged holomorphic map \((A,u)\)  is given by
\( E(A,u) = {{2}}\int _\Sigma \left(| {\operatorname{d}}_A u |^2 + |F_A|^2 + |u^*P(\Phi )|^2 \right)  \operatorname{Vol}_\Sigma .\) 
The equivariant symplectic area of a pair \((A,u)\)  is the
pairing of the homology class \(\deg (u)\)  with the class \([\omega _G =\omega + \Phi ] \in H^2(X_G)\) ,
\( D(A,u) = (\deg (u), [\omega _G]) = ([\Sigma ], u^* [\omega _G] ) .\) 
Lemma 4.0.24 Suppose \( \operatorname{Vol}_\Sigma \)  is the Kähler form
for the metric on \(\Sigma \) . The energy and equivariant area are
related by
\( E(A,u) = D(A,u) + \int _\Sigma \left( | \overline{\partial }_A u |^2 + {{2}}| F_A +  \operatorname{Vol}_\Sigma u^* P(\Phi ) |^2 \right)  \operatorname{Vol}_\Sigma .\) 

See [26]}.
In particular, for any symplectic vortex the energy and action are
equal. Let \(M(P,X)\)  denote the moduli space of vortices
\( M(P,X) := \mathcal {H}(P,X) /\hspace{-2.97498pt}/\mathcal {G}(P) = \lbrace  F_A +  \operatorname{Vol}_{\Sigma } u^* P(\Phi ) =0 \rbrace /\mathcal {G}(P) .\) 
Let \(M_n(P,X)\)  denote the moduli space of \(n\) -marked vortices, up to
gauge transformation, and
\(M_n(\Sigma ,X) = \bigcup _{P \rightarrow \Sigma } M_n(P,X) \) 
the union over types of bundles \(P\) . Clearly, \(M_n(\Sigma ,X)\)  is
homeomorphic to the product \(M(\Sigma ,X) \times M_n(\Sigma ) \)  where
\(M(\Sigma ,X) := M_0(\Sigma )\)  and \(M_n(\Sigma )\)  denotes the
configuration space of \(n\) -tuples of distinct points on \(\Sigma \) .
We wish to study families and deformations of symplectic vortices.
For families with smooth domain, the definitions are straightforward:
Definition 4.0.25 
A smooth family of vortices on a principal \(G\) -bundle \(P\)  on
\(\Sigma \)  over a parameter space \(S\)  consists of a family of
connections depending smoothly on \(s \in S\) , that is, a smooth map
\(A_S : S \times P \rightarrow T^*P \otimes \mathfrak {g}\)  on \(P\)  such that the
restriction \(A_s\)  of \(A_S\)  to any \( \lbrace s \rbrace  \times P\)  is a connection,
together with a smooth family of (pseudo)holomorphic sections \(u_S =(u_s)_{s \in S}\) , such that each pair \((A_s,u_s), s \in S\)  is a
symplectic vortex. A deformation of \((A,u)\)  is a germ of a
smooth family \((A_S,u_S)\)  together with an isomorphism (gauge
transformation) relating \((A_0,u_0)\)  with \((A,u)\) . A deformation is
universal if it satisfies the condition as in Definition
REF , and strongly universal if it satisfies the
conditions in Definition REF .

We define a linearized operator associated to a vortex as
follows. Define
\( {\operatorname{d}}_{A,u}: \Omega ^1(\Sigma ,P(\mathfrak {g})) \oplus \Omega ^0(\Sigma ,u^*TP(X) ) \rightarrow \Omega ^2(\Sigma ,P(\mathfrak {g})) \) 
\( {\operatorname{d}}_{A,u}(a,\xi ) := {\operatorname{d}}_A a +  \operatorname{Vol}_{\Sigma } u^* L_\xi P(\Phi ).\) 
Here \(L_\xi P(\Phi )\)  denotes the derivative of \(P(\Phi )\)  with respect
to the vector field generated by \(\xi \) , and \(u^* L_\xi P(\Phi )\)  its
evaluation at \(u\) . Define an operator
\( {\operatorname{d}}_{A,u}^*: \Omega ^1(\Sigma ,P(\mathfrak {g})) \oplus \Omega ^0(\Sigma ,u^* TP(X) )\rightarrow \Omega ^0(\Sigma ,P(\mathfrak {g})) \) 
\( {\operatorname{d}}_{A,u}^*(a,\xi ) = {\operatorname{d}}_A^*a + u^* L_{J \xi } P(\Phi ) .\) 
(This is not the adjoint of operator in (REF ), but rather
defined by analogy with the case \(X\)  trivial.) It is shown in
[23]} that if \((A,u)\)  is stable then the set
\( W_{A,u} = \lbrace  (A + a, \exp _u(\xi )), (a,\xi ) \in  \operatorname{ker}{\operatorname{d}}_{A,u}^*\rbrace  \) 
is a slice for the gauge group action near \((A,u)\) . Define
\( \mathcal {F}_{A,u} : \Omega ^1(\Sigma ,P(\mathfrak {g})) \oplus \Omega ^0(\Sigma ,u^* T^{\operatorname{vert}} P(X)) \\ \rightarrow (\Omega ^0 \oplus \Omega ^2)(\Sigma ,P(\mathfrak {g})) \oplus \Omega ^{0,1}(\Sigma ,u^*T^{\operatorname{vert}} P(X)) \\(a,\xi ) \mapsto \left( F_{A + a} +  \operatorname{Vol}_\Sigma \exp _u(\xi )^* P(\Phi ), {\operatorname{d}}_{A,u}^* (a,\xi ), \Psi _u(\xi )^{-1}\overline{\partial }_{A + a} \exp _u(\xi ) \right) .\) 
Let
\(\Omega ^1(\Sigma ,P(\mathfrak {g})) \rightarrow \Omega ^1(\Sigma ,u^*T^{\operatorname{vert}}P(X)), \ \ \ a \mapsto a_X \) 
denote the map induced by the infinitesimal action. The linearization
of the last component (REF ) is
\( D_{A,u} (a,\xi ) = (\nabla _A \xi )^{0,1} + {{2}}J_u (\nabla _\xi J )_u \partial _A u+ a_X^{0,1}.\) 
Here \(0,1\)  denotes projection on the \(0,1\) -component. The linearized
operator for a vortex \((A,u)\)  is the operator
\( \tilde{D}_{A,u} = ({\operatorname{d}}_{A,u}, {\operatorname{d}}_{A,u}^*, D_{A,u} ):\Omega ^1(\Sigma ,P(\mathfrak {g})) \oplus \Omega ^0(\Sigma , u^* T^{\operatorname{vert}}P(X))\\ \rightarrow (\Omega ^0 \oplus \Omega ^2)(\Sigma ,P(\mathfrak {g})) \oplus \Omega ^{0,1}(\Sigma , u^* T^{\operatorname{vert}}P(X))\) 
A vortex \((A,u)\)  is regular if the operator \(\tilde{D}_{A,u}\)  is
surjective. A marked vortex \((A,u,{z})\)  is regular if the
underlying unmarked vortex is regular. The space of
infinitesimal deformations of \((A,u)\)  is
\( \operatorname{Def}(A,u) :=  \operatorname{ker}(\tilde{D}_{A,u}) .\)
Theorem 4.0.26 
Any regular vortex with smooth domain \((A,u)\)  has a strongly universal
smooth deformation if and only if it is stable.

Give the spaces of connections and sections the structure of Banach
manifolds by taking completions with respect to Sobolev norms \(1,p\) 
for 1-forms, and \(0,p\)  for 0 and 2-forms. For \(p>2\) , the map
\(\mathcal {F}_ {A,u}\)  is a smooth map of Banach spaces.
\(\mathcal {F}_{A,u} : \Omega ^1(\Sigma ,P(\mathfrak {g}))_{1,p} \oplus \Omega ^0(\Sigma ,u^*T^{\operatorname{vert}} P(X))_{1,p} \\ \rightarrow (\Omega ^0 \oplus \Omega ^2)(\Sigma ,P(\mathfrak {g}))_{0,p} \oplus \Omega ^{0,1}(\Sigma ,u^*T^{\operatorname{vert}} P(X))_{0,p} \) 
equivariant for the action of the group \(\mathcal {G}(P)_{2,p}\)  of gauge
transformations of class \(2,p\) . Suppose that \((A,u)\)  is regular and
stable. By the implicit function theorem, there is a local
homeomorphism
\(  \operatorname{ker}(\tilde{D}_{A,u}) \rightarrow \left\lbrace  \begin{array}{c} F_{A+ a} +  \operatorname{Vol}_\Sigma (\exp _u(\xi ))^* P(\Phi ) = 0 \\ \overline{\partial }_{A + a}(\exp _u(\xi )) = 0 \\ \ {\operatorname{d}}_{A,u}^*(a,\xi ) = 0 \end{array} \right\rbrace .\) 
This gives rise to a family \((A_S,u_S) \rightarrow S\)  over a neighborhood \(S\) 
of 0 in \( \operatorname{ker}(\tilde{D}_{A,u})\) . By [23]},
\((A_S,u_S)\)  is a smooth family, assuming \((A,u)\)  is smooth. Given any
other family \((A_{S^{\prime }}^{\prime },u_{S^{\prime }}^{\prime }) \rightarrow S^{\prime }\)  of stable vortices with
\((A^{\prime }_0,u_0^{\prime }) = (A,u)\) , the implicit function theorem provides a smooth
map \(S^{\prime } \rightarrow S\)  so that \((A^{\prime }_{S^{\prime }},u^{\prime }_{S^{\prime }})\)  is obtained from \((A,u)\)  by
pull-back. The first property of the universal deformation is a
consequence of the slice condition; the second property follows from
the fact that the projection \( \operatorname{ker}(\tilde{D}_{A,u}) \rightarrow  \operatorname{ker}(\tilde{D}_{A_s,u_s})\)  is an isomorphism for sufficiently small \(s\) .
Let \(M^{\operatorname{reg}}_n(\Sigma ,X)\)  denote the moduli space of regular,
stable \(n\) -marked symplectic vortices from \(\Sigma \)  to \(X\) . We
denote by \((c_1^G(TX),d)\)  the pairing of \(d\)  with the first Chern
class \(c_1^G( P(TX) \rightarrow P(X))\)
Theorem 4.0.27  Let \(\Sigma ,X,J\)  be as above.
\(M^{\operatorname{reg}}_n(\Sigma ,X)\)  has the structure of a smooth orbifold with
tangent space at \([A,u]\)  isomorphic to \(\operatorname{Def}(A,u)\) , and dimension of
the component of homology class \(d \in H_2^G(X)\)  is given by
\(\dim (M^{\operatorname{reg}}_n(\Sigma ,X,d))= (1-g)(\dim (X) - 2\dim (G)) +2((c_1^G(TX),d) + n) .\) 

Charts for \(M^{{\operatorname{reg}}}_n(\Sigma ,X)\)  are provided by the strongly
universal deformations. The dimension of the tangent space at \([A,u]\) 
is given by the index of the linearized operator \(\tilde{D}_{A,u}\) , which
deforms via Fredholm operators to the sum of the operator \({\operatorname{d}}_A \oplus {\operatorname{d}}_A^*\)  for the connection, which has index \(2\dim (G)(g-1)\) , and the
linearized Cauchy-Riemann operator on the nodal curve, which has index
\((1-g)\dim (X) + 2n + 2(c_1^G(TX),d)\)  by Riemann-Roch, if \((A,u)\)  has
equivariant homology class \(d\)  (which determines the first Chern class
of \(P\)  by projection.)

Polystable vortices
The moduli space of symplectic vortices admits a compactification
which allows bubbling of the section in the fibers.
Definition 4.1.1 
A nodal gauged marked holomorphic map from \(\Sigma \)  to \(X\) 
consists of a datum \((\hat{\Sigma },P,A,{u},{z})\)  where \(P \rightarrow \Sigma \)  is a principal \(G\) -bundle, \(A \in A(P)\)  is a connection,
\(\hat{\Sigma }\)  is a marked nodal curve, \(v: \hat{\Sigma } \rightarrow \Sigma \) 
is a holomorphic map of degree \([\Sigma ]\) , and \({u}: \hat{\Sigma }\rightarrow P(X)\)  is a \(J_A\) -holomorphic map from a nodal curve \(\hat{\Sigma }\) 
such that \(\pi \circ {u}\)  has class \([\Sigma ]\) . In other words,

\(\hat{\Sigma }\)  is a connected nodal complex curve consisting of
a principal component \(\Sigma _0\)  equipped with an isomorphism
with \(\Sigma \)  together with a number of projective lines
\(\Sigma _1,\ldots ,\Sigma _k\) . We denote by \(w_1^\pm ,\ldots ,w_m^\pm \) 
the nodes. For each \(i = 1,\ldots , k\) , we denote by \(w_i^0 \in \Sigma _0\)  the attaching point to the principal component.

\((A,u) \in \mathcal {H}(P,X)\) 
is a gauged holomorphic map from \(\Sigma \)  to \(X\) ;

for each non-principal component \(\Sigma _i\) , a holomorphic map
\(u_i: \Sigma _i \rightarrow P(X)_{w_i^0}\) ;

\({z} = (z_1,\ldots ,z_n) \in \hat{\Sigma }\)  are distinct, smooth
points of \(\hat{\Sigma }\) .


Let \(\mathcal {H}(\hat{\Sigma },P(X))\)  denote the space of nodal gauged
holomorphic sections with domain \(\hat{\Sigma }\)  and bundle \(P\) . The
group of gauge transformations \(\mathcal {G}(P)\)  acts on \(\mathcal {H}(\hat{\Sigma },P(X))\) 
by \( g (A,{u}) = (g^* A, g \circ {u}) .\)  The generating vector
field for \(\zeta \in \Omega ^0(\Sigma ,P(\mathfrak {g}))\)  acting on
\(\mathcal {H}(\hat{\Sigma },P(X))\)  at \((\hat{\Sigma },A,{u})\)  is the tuple
given by
\( \zeta _{\mathcal {H}(\hat{\Sigma },P(X))}(\hat{\Sigma },A,{u}) = ({\operatorname{d}}_A \zeta ,u_0^* P(\zeta _X), (u_i^* P(\zeta _X(w_i^0)))_{i=1}^k )\) 
in \( \Omega ^1(\Sigma ,P(\mathfrak {g})) \oplus \Omega ^0(\hat{\Sigma },{u}^*T^{\operatorname{vert}} P(X))\) . Here \(P(\zeta _X) \in \Omega ^0(\Sigma , P(\operatorname{Vect}(X)))\)  is the fiber-wise vector field generated by \(\zeta \)  and
\(u_0^* P(\zeta ) \in T^{\operatorname{vert}} P(X)\)  is the evaluation at \(u_0\) .
Similarly for the bubble components \(u_1,\ldots , u_k\)  in the fibers
\(P(X)_{w^0_1},\ldots , P(X)_{w^0_k}\) . A slice is given by taking the
perpendicular to the tangent spaces to the \(\mathcal {G}(P)\) -orbits. We will
assume for simplicity that the stabilizer of the \(\mathcal {G}(P)\)  action on the
principal component is finite, so that a slice is given locally by the
kernel of \({\operatorname{d}}_{A,u_0}^*\) , that is, the Coulomb gauge condition on the
principal component. The implicit function theorem shows that any
nearby pair \((A_1,{u}_1)\)  is complex gauge equivalent to a pair of
the form \((A + a, \exp _{{u}}({v}))\)  with \((a,{v}) \in  \operatorname{ker}{\operatorname{d}}_{A,u_0}^*\) .
Definition 4.1.2 A nodal vortex is a stable nodal gauged holomorphic map such
that the principal component is an vortex. A nodal vortex
\((\hat{\Sigma },A,{u},{z})\)  is polystable if each sphere
bubble \(\Sigma _i\)  on which \(u_i\)  is constant has at least three marked
or singular points, and stable if it has finite automorphism
group.
An isomorphism of nodal vortices
\((\hat{\Sigma },A,{u},{z}),(\hat{\Sigma }^{\prime },A^{\prime },{u}^{\prime },{z}^{\prime })\) 
consists of an automorphism of the domain, acting trivially on the
principal component, and a corresponding automorphism of the
principal bundle mapping \((A,{u}) \)  to \((A^{\prime },{u}^{\prime })\)  and mapping
the markings \({z}\)  to \({z}^{\prime }\) .
For any nodal section \({u}:\hat{\Sigma } \rightarrow P(X)\) , the homology
class of \({u}\)  is defined as the sum of the homology class \(d_0\in H_2^G(X,\mathbb {Z})\)  of the principal component \(u_0\)  and the homology
classes \(d_i \in H_2(X,\mathbb {Z}), i = 1,\ldots ,k\)  of the sphere bubbles,
using the inclusion \(H_2(X,\mathbb {Z}) \rightarrow H_2^G(X,\mathbb {Z})\)  given by equivariant
formality.
The combinatorial type \(\Gamma (\hat{\Sigma },A,{u},{z})\)  of
a gauged nodal map is a rooted graph whose vertices represent the
components of \(\hat{\Sigma }\) , whose finite edges represent the nodes,
semi-infinite edges represent the markings, and whose root vertex
represents the principal component.

Note that there is no condition for points on the principal
component. In particular, nodal gauged holomorphic maps with no
markings can be polystable. The term polystable is borrowed
from the vector bundle case. In that situation, a bundle is stable if it is flat and has only central automorphisms and
polystable if it is a direct sum of stable bundles of the same slope.
Any flat bundle is automatically polystable; a bundle is semistable if it is grade equivalent to a polystable bundle. In
particular, the moduli space of stable bundles is definitely not
compact, and we feel that the vortex terminology should include this
fact as a special case.
From now on, we fix the bundle \(P\) .
Definition 4.1.3 Let \(X\)  as above.
A smooth family of fixed type of nodal vortices to \(X\)  consists
of a smooth family \(\hat{Sigma}_S \rightarrow S\)  of nodal curves of fixed
type, a smooth family of holomorphic maps \(v_S: \hat{\Sigma }_S \rightarrow \Sigma \)  of class \([\Sigma ]\) , a smooth family \({u}_S :\hat{\Sigma }_S \rightarrow P(X)\)  of maps, and a smooth family \(A_S : S \times P \rightarrow T^* P\)  of connections over \(S\) . A smooth deformation of a
nodal vortex \((A,\hat{\Sigma },{u},{z})\)  of fixed type consists
of a germ of a smooth family
\((A_S,\hat{\Sigma }_S,{u}_S,{z}_S)\)  of nodal vortices of fixed
type together with an identification \(\iota \)  of of the central fiber
with \((A,\hat{\Sigma },{u},{z})\) . A stratified-smooth
family of marked nodal symplectic vortices is a datum
\((\hat{\Sigma }_S,A_S,{u}_S,{z}_S)\)  consisting of a
stratified-smooth family \(\hat{\Sigma }_S \rightarrow S\)  of nodal curves, a
stratified-smooth family of holomorphic maps \(v: \hat{\Sigma }_S \rightarrow \Sigma \)  of class \([\Sigma ]\) , a stratified-smooth family \(A_S\)  of
connections on \(P\) , a stratified-smooth family of maps \({u}_S:\hat{\Sigma }_S \rightarrow P(X)\) ; such that each triple
\((\hat{\Sigma }_s,A_s,{u}_s,{z}_s)\)  is a marked nodal
symplectic vortex. A family of polystable symplectic vortices
is a family of marked nodal symplectic vortices such that any fiber is
polystable.

A smooth vector bundle \({E} \rightarrow \hat{\Sigma }\)  is a
collection of smooth vector bundles \(E_i\)  over the components
\(\Sigma _i\)  of \(\hat{\Sigma }\) , equipped with identifications of the
fibers at nodal points \(E_{i^+(w^+_k)} \rightarrow E_{i^-(w^-_k)}, k =1,\ldots , m\) . We denote by \(\Omega (\hat{\Sigma },{E})\)  the sum
over components, \( \Omega (\hat{\Sigma },{E}) = \bigoplus _{i=1}^k\Omega (\Sigma _i, E_i) \)  where \(E_i = {E} | \Sigma _i .\)
Definition 4.1.4 For a polystable vortex
\((\hat{\Sigma },A,{u})\) , let \(\tilde{D}_{A,{u}}\)  denote the linearized operator
\(\Omega ^1({\Sigma },P(\mathfrak {g})) \oplus \Omega ^0(\hat{\Sigma },{u}^*T^{\operatorname{vert}} P(X))\\ \rightarrow (\Omega ^{0}\oplus \Omega ^2)({\Sigma },P(\mathfrak {g})) \oplus \Omega ^{0,1}(\hat{\Sigma },{u}^*T^{\operatorname{vert}} P(X)) \oplus \bigoplus _{k=1}^m T_{{u}(w_k^\pm )}^{\operatorname{vert}} P(X)\) 
given by the linearized vortex operator \(({\operatorname{d}}_{A,u_0}, D_{A,u_0})\)  on
the principal component, the linearized Cauchy-Riemann operator
\(\tilde{D}_{u_i}\)  on the bubbles, the slice operator \({\operatorname{d}}_{A,u_0}^*\) , and
the difference operator on the fibers over the nodes
\( \Omega ^0(\hat{\Sigma },{u}^* T^{\operatorname{vert}} P(X)) \rightarrow \bigoplus _{i=1}^m T_{{u}(w_i^\pm )}^{\operatorname{vert}} P(X), \quad {\xi }\mapsto ( {\xi }(w_i^+) - {\xi }(w_i^-))_{i=1}^m .\) 
\((A,{u})\)  is regular if \(\tilde{D}_{A,{u}}\)  is surjective.
The space of infinitesimal deformations of \(A,{u}\)  of fixed type is
\( \operatorname{Def}_\Gamma (A,{u}) :=  \operatorname{ker}\tilde{D}_{A,{u}}/ \operatorname{aut} (\hat{\Sigma })\) 
where \( \operatorname{aut} (\hat{\Sigma })\)  denotes the group of infinitesimal
automorphisms acting trivially on the principal component. The space
of infinitesimal deformations of \(A,{u}\)  is
\( \operatorname{Def}(A,{u}) := \operatorname{Def}_\Gamma (A,{u}) \oplus \bigoplus _{i=1}^mT_{w_j^+} \hat{\Sigma } \otimes T_{w_j^-} \hat{\Sigma } \) 
consisting of a deformation of fixed type together with a collection
of gluing parameters.


Constructing deformations of symplectic vortices
First we construct deformations of fixed type.
Theorem 4.2.1 
A regular polystable vortex has a strongly universal smooth
deformation of fixed type if and only if it is stable.

The proof is by the implicit function theorem applied to the
map
\( \mathcal {F}_{A,{u}} (a,{\xi }) = (F_{A + a} +  \operatorname{Vol}_\Sigma (\exp _{u_0}(\xi _0))^* P(\Phi ), {\operatorname{d}}_{A,u_0}^* (a,{\xi }), \\\Psi _{u_0}(\xi _0)^{-1} \overline{\partial }_A u_0, (\Psi _{u_i}(\xi _i)^{-1} \overline{\partial }u_i)_{i=1}^k, ({\xi }(w_i^+) - {\xi }(w_i^-))_{i=1}^m )\) 
whose linearization is \(\tilde{D}_{A,{u}}\) . The proof is left to the
reader. We denote by \(M_{n,\Gamma }(\Sigma ,X,d)\)  of the moduli space
of isomorphism classes of polystable vortices of combinatorial type
\(\Gamma \)  of homology class \(d \in H_2^G(X,\mathbb {Z})\) , and
\(M^{{\operatorname{reg}}}_{n,\Gamma }(\Sigma ,X,d)\)  the regular locus.
Corollary 4.2.2 
\(M_{n,\Gamma }^{\operatorname{reg}}(\Sigma ,X,d)\)  has the structure of a smooth orbifold
of dimension given, if \(\Sigma \)  is connected, by \((1 - g)(\dim (X) -2\dim (G)) + 2( n + (c_1^G(TX),d) - m)\)  where \(m\)  is the number of
nodes.

We now prove that a regular stable symplectic vortex from \(\Sigma \)  to
\(X\)  admits a strongly universal stratified-smooth deformation if it is
strongly stable, that is, Theorem REF . We explain the
construction for a single bubble only, so that \(\hat{\Sigma }\)  is the
union of a principal component \(\Sigma _+ = \Sigma \)  and a holomorphic
sphere \(\Sigma _-\) , attached by a single pair \(w_\pm \)  of nodes. We
denote by \((A,u_+)\)  the restriction to the principal component and by
\(u_-\)  the bubble, so that \(x := u_+(w_+) = u_-(w_-)\)  and \({u} =(u_+,u_-)\) . We choose a local coordinate near \(w\) , equivariant for
the action of the automorphism group \( \operatorname{Aut} (A,{u})\)  in the sense
that \( \operatorname{Aut} (A,{u})\)  acts on the local coordinate by multiplication
by roots of unity. The construction depends on the following choices:
Definition 4.2.3 A gluing datum for \((\hat{\Sigma },A,{u})\)  consists of

neighborhoods \(U_\pm \)  of the nodes \(w_\pm \) ;

local coordinates \({\kappa } = (\kappa _+,\kappa _-)\)  on
\(U_\pm \) ;

a trivialization \( P |_{U_0} \rightarrow G \times U_0 \)  on \(U_0\) ;

a gluing parameter \(\delta \) ;

an annulus parameter \(\rho \)  ;

a cutoff function \(\alpha \)  as in (REF ).


Step 1: Approximate Solution
Given a nodal vortex \((A,u)\)  as above and a gluing datum we wish to
define an approximate solution to the vortex equations \((A,u^\delta )\) . Let \(\exp _x: T_x X \rightarrow X \)  denote the exponential map
defined by the metric on \(X\) . Define sections
\(\xi _\pm : U_\pm \rightarrow T_x X, \quad u(z)=\exp _{x}(\xi _\pm (z)) .\) 
Let \(\hat{\Sigma }^\delta \)  denote the surface obtained by gluing; since
the bubble is genus zero, this surface is isomorphic to \(\Sigma \)  but
not canonically. Define the pre-glued section \({u}^\delta :\hat{\Sigma }^\delta \rightarrow P(X)\) ,
\({u}^\delta (z) = \exp _{x} \left( \alpha (|\kappa _+(z)|/ \rho |\delta |^{1/2}) (\xi _+(z) - \xi _\pm (w^\pm )) \right. +\\ \left. \alpha (|\kappa _-(z)|/ \rho |\delta |^{1/2} ) (\xi _-(z) - \xi _\pm (w^\pm )) +\xi _\pm (w^\pm ) \right)\) 
for \(| \kappa _\pm (z) | \le 2|\delta |^{1/2} \rho ^2\) ; elsewhere let
\({u}^\delta (z) = {u}(z)\) , using the identification of \(\Sigma \) 
with \(\hat{\Sigma }\)  away from the gluing region. We do not modify \(A\) 
in the bubble region; this is because after re-scaling the connection
on the bubble is already close to the trivial connection.
The pair \((A,{u}^\delta )\)  is an approximate solution to the vortex
equations in the following Sobolev norms. Let
\(\Omega (\Sigma ^\delta ,P^\delta (\mathfrak {g}))_{k,p}\)  denote the
\(W^{k,p}\) -completion using the standard metric. Let \(g^\delta \)  denote
the \(C^0\)  metric on the glued surface in (REF ). Let
\(\Omega ^{0,1}(\Sigma ^\delta , u^{\delta ,*}T^{\operatorname{vert}}P(X))_{k,p,\delta } \) . Let
\( \mathcal {H}_{\delta } := \Omega ^2(\Sigma ,P(\mathfrak {g}))_{{0,3}} \oplus \Omega ^0(\Sigma ,P(\mathfrak {g}))_{0,3} \oplus \Omega ^{0,1}(\Sigma ^\delta , u^{\delta ,*}T^{\operatorname{vert}}P(X))_{{0,3},\delta }\) 
with norm
\( \Vert (\phi ,\psi ,\eta ) \Vert _{{\delta }}^2 = \Vert \phi \Vert _{{0,3}}^2 + \Vert \psi \Vert _{0,3}^2 + \Vert \eta \Vert _{0,3,\delta }^2 .\) 
Let
\( \mathcal {I}_{\delta } := \Omega ^1(\Sigma , P(\mathfrak {g}))_{{1,3}} \oplus \Omega ^0(\Sigma ^\delta , u^{\delta ,*}T^{\operatorname{vert}}P(X))_{{1,3},\delta } \) 
with norm
\( \Vert (a, \xi ) \Vert _{{\delta }}^2 = \Vert a \Vert _{{1,3}}^2 + \Vert \eta \Vert _{{1,3},\delta }^2 .\) 
Locally the moduli space of polystable vortices is in bijection with
the zero set of the map
\( \mathcal {F}_{A,{u}}^{{\delta }} : \mathcal {I}_{{\delta }} \rightarrow \mathcal {H}_{{\delta }} \\(a,\xi ) \mapsto \left( F_{A + a} +  \operatorname{Vol}_\Sigma \exp _{{u}^{{\delta }}}(\xi )^* P(\Phi ), {\operatorname{d}}_{A,u_+}^* (a,\xi ),\Psi _{{u}^{{\delta }}}(\xi )^{-1} \overline{\partial }_{A + a}\exp _{{u}^{{\delta }}}(\xi ) \right) .\) 
Here the second map enforces a slice condition. That
\(\mathcal {F}_{A,{u}}^{{\delta }}\)  is well-defined follows from Sobolev
embedding: In particular, there is a \(\delta \) -uniform embedding
\( \Omega ^0(\Sigma ^\delta , u^{\delta ,*}T^{\operatorname{vert}}P(X))_{{1,3},\delta } \rightarrow \Omega ^0(\Sigma ^\delta ,u^{\delta ,*} T^{\operatorname{vert}}P(X))_{0,\infty }, \) 
since the dimensions of the cone in the cone condition [7]} are uniformly bounded in \(\delta \) , and the metric
uniformly comparable to the flat metric.
Lemma 4.2.4 
Let \((A,{u})\)  be a symplectic vortex on a nodal curve with a
single node \(w = (w^+,w^-)\) . There exist constants \(c_0,c_1 > 0\)  such
that if
\(|\delta | < c_1, \rho > 1/c_1 \)  and\( |\delta | \rho ^4 < c_1 \) 
then the pair \((A,{u}^\delta ) \in \mathcal {A}(P) \times \Gamma (P(X))\) 
satisfies
\(\Vert \mathcal {F}_{A,{u}}^{{\delta }}(0,0) \Vert \le c_{0} |\delta |^{1/3}\rho ^{2/3}.\) 

The expression \(\overline{\partial }_{A} {u}^\delta \)  can be expressed as a sum of
terms involving derivatives of the cutoff function \(\alpha \) , terms
involving derivatives of \(\xi _j\) , and terms involving the connection
\(A\)  on the bubble region. The derivative of \(\alpha \)  is bounded by \(C/ \rho | \delta |^{1/2}\) , while the norm of \(\xi _j\)  is bounded by
\(C\rho |\delta |^{1/2}\)  on the gluing region. Hence the term
involving the derivative of \(\alpha \)  is bounded and supported on a
region of area less than \( C|\delta | \rho ^2\) . In the given
trivialization we have
\( \overline{\partial }_A u = \overline{\partial }u + A_X^{0,1}(u) \) 
where \(A^{0,1}\)  is the \(0,1\) -form defined by \(A \in \Omega ^1(B_R,\mathfrak {g})\) 
is the connection 1-form in the local trivialization and
\(A_X^{0,1}(u)\)  is the corresponding form with values in \(T^{\operatorname{vert}}P(X) \otimes _\mathbb {R}. We have\)\( \Vert A_X^{0,1}(u) \Vert _{0,3,\delta } \le \Vert A_X^{0,1}(u)\Vert _{0,3} \)\(since \) p 2\(; for \) p = 2\( the \) W0,3,\( and \) W0,3\( normsare the same, by conformal invariance; for \) p > 2\( the \) 0,3\(-norm isstrictly greater. Hence\)\( \Vert \overline{\partial }_{A} u^\delta \Vert _{0,3,\delta } \le C\max (|\delta |^{1/3} \rho ^{2/3},|\delta |). \)\(The moment map term \) FA + (u)* P()  Vol\(vanishes except on \) |+| |1/2|\(, where it isuniformly bounded. Hence for \)\( small\)\( \Vert F_A + ({u}^\delta )^* P(\Phi )  \operatorname{Vol}_\Sigma \Vert _{0,3} \le C \rho ^{2/3} |\delta |^{1/3} .\)\(The statement of the lemma follows.\)
We also wish to perform the gluing construction in families, that is,
for each nearby vortex and gluing parameter we wish to find a solution
to the vortex equations on \(\Sigma ^\delta \) . Define
\( \mathcal {F}_{A,{u}}^{D,\delta } : \operatorname{Def}_\Gamma (A,{u}) \times \mathcal {I}_{{\delta }} \rightarrow \mathcal {H}_{{\delta }}, \quad (a,{\xi },a_1,\xi _1) \mapsto \\ \left( F_{A} + a_0+ a_1 +  \operatorname{Vol}_\Sigma \exp _{{u}^{{\delta }}}({\xi }_0^\delta +\xi _1)^* P(\Phi ), {\operatorname{d}}_{A,{u}^\delta }^* (a_0 +a_1,{\xi }_0^\delta + \xi ),\right. \\ \left. \Psi _{{u}^{{\delta }}}({\xi }_0^\delta + \xi _1)^{-1}\overline{\partial }_{A + a_0 + a_1}\exp _{{u}^{{\delta }}}({\xi }_0^\delta + \xi _1)\right) \) 
The following is proved in the same way as Lemma REF  and left
to the reader:
Lemma 4.2.5 
Let \((A,{u})\)  be as above. There exist constants \(c_0,c_1 > 0\) 
such that if \(|\delta | < c_1, \rho > 1/c_1 \) , \(\Vert (a,\xi ) \Vert \le c_1\)  and \(|\delta | \rho ^4 < c_1 \)  then
\(\Vert \mathcal {F}_{A,{u}}^{D,{\delta }}(a,\xi ,0,0) \Vert \le c_{0}|\delta |^{1/3} \rho ^{2/3}.\) 

Step 2: Uniformly bounded right inverse
In preparation for the construction of the uniformly bounded right
inverse of \(\tilde{D}_\delta \)  we define the intermediate family
\((A,{u}_0^\delta )\)  of gauged holomorphic maps on the nodal curve
\(\hat{\Sigma }\)  is the family defined by the equations (REF ),
using the identification of \(\hat{\Sigma }\)  and \(\hat{\Sigma }^\delta \) 
away from the gluing region. Thus \(u_0^\delta \)  is constant in a
neighborhood of the node \(w^\pm \) . We identify \(({u}_0^\delta )^*T^{\operatorname{vert}} P(X)\)  with \({u}^* T^{\operatorname{vert}} P(X)\)  by geodesic
parallel transport.
Lemma 4.2.6 
The operator \(\tilde{D}_{A,{u}_0^\delta }\) 
converges in the operator norm to \(\tilde{D}_{A,{u}}\)  as \(\delta \rightarrow 0, \rho \rightarrow \infty , \delta \rho ^4 \rightarrow 0\) .

The section \({u}_0^\delta \)  converges in the \(W^{1,3}\)  norm to \(u\) 
as \(\rho ^2 |\delta |^{1/2} \rightarrow 0\) . It follows that the operator \(\xi \mapsto  \operatorname{Vol}_\Sigma (u^\delta _0)^* L_\xi P(\Phi )\)  converges to \(\xi \mapsto  \operatorname{Vol}_\Sigma u^* L_\xi P(\Phi )\) . Hence \({\operatorname{d}}_{A,u_0^\delta ,\epsilon }\)  converges to \({\operatorname{d}}_{A,u,\epsilon }\) , and similarly
for \({\operatorname{d}}_{A,u_0,\epsilon }^*\) . The operator \( D_{A,{u}_0^\delta }\) 
converges to \(D_{A,{u}}\) , as in Lemma REF .
Proposition 4.2.7 
Let \((A,{u})\)  be a nodal vortex, and \((A,{u}^\delta )\)  the
approximate solution constructed above. There exist constants
\(c,C > 0\)  such that if \(| \delta | < c\)  then there exists an
approximate right inverse \(T_{\delta }\)  of the parametrized linear
operator \(\tilde{D}_{\delta } := \tilde{D}_{A,{u}^\delta }\)  that is,
a map \(T_{\delta } : \mathcal {H}_{\delta } \rightarrow \mathcal {I}_{\delta } \) 
such that
\( \Vert (\tilde{D}_{\delta } T_{\delta } - I) \eta \Vert _{\delta } \le {{2}}\Vert \eta \Vert _{\delta }, \quad \Vert T_{{\delta }} \eta \Vert _{\delta } \le C \Vert \eta \Vert _{\delta } .\) 

Given such an approximate inverse, we obtain a uniformly bounded right
inverse \(Q_{\delta }\)  to \(\tilde{D}_{\delta }\)  by the formula
\( Q_{\delta } = T_{\delta } (\tilde{D}_{\delta } T_{\delta })^{-1} = \sum _{k \ge 0}T_{\delta } (\tilde{D}_{\delta } T_{\delta } - I )^k .\) 
By the regularity assumption, \(\tilde{D}_{A,{u}} \)  is surjective when
restricted to the space of vectors \((a,{\xi })\)  such that \(\xi _0(0)= \xi _1(\infty )\) . By Lemma REF , \(\tilde{D}^0_\delta :=\tilde{D}_{A,{u}_0^\delta }\)  is surjective for sufficiently small
\(\rho ,\delta \) , when restricted to the same space. The approximate
right inverse is constructed by composing a cutoff operator
\(K_\delta \) , right inverse \(Q_\delta \) , and gluing operator \(R_\delta \) ,
as follows. (In other words, the right inverse is defined by
truncating the given functions, applying the right inverse for the
linearized operator on the nodal curve, and then gluing back together
using cutoff functions again.) For simplicity we assume that there is
a single node. Define the cutoff operator
\(K_{\delta }:\Omega ^{0,1}(\hat{\Sigma }^\delta ,{u}^{\delta ,*} T^{\operatorname{vert}}P(X))_{0,3,\delta }\rightarrow \Omega ^{0,1}(\hat{\Sigma },({u}^\delta _0)^{*}T^{\operatorname{vert}}P(X))_{0,3} \\K_\delta (\eta ) ={\left\lbrace \begin{array}{ll} 0 & \kappa _\pm ( z) \in B_{| \delta |^{{{2}}}} (0) \\\eta & \text{otherwise}\end{array}\right.} .\) 
Then \( \Vert K_\delta (\eta ) \Vert _{0,3} \le \Vert \eta \Vert _{0,3,\delta } .\)  Define the gluing operator
\( R_{\delta }:\Omega ^0(\hat{\Sigma },({u}^\delta _0)^{*} TX)_{{1,3}} \rightarrow \Omega ^0(\hat{\Sigma }^{\delta },u^{\delta ,*}T^{\operatorname{vert}}P(X))_{{1,3},\delta }\) 
as follows. Let \(\hat{\Sigma }_\pm ^*\)  denote the complements of small
balls around the nodes \( \hat{\Sigma }_\pm ^* = \Sigma _\pm - B_{\rho ^2 |\delta |^{1/2}} (w^\pm )\) . Let \(\pi _\pm : \hat{\Sigma }_\pm ^* \rightarrow \hat{\Sigma }^\delta \)  denote the inclusions. These induce maps of
sections with compact support in \(\hat{\Sigma }_\pm ^*\) , \( \pi _{\pm ,*}:\, \Omega ^0_c( \hat{\Sigma }_\pm ^*,u_\pm ^* TX) \rightarrow \Omega ^0(\hat{\Sigma }^\delta , u^{\delta ,*} TX ) .\)  Define \(R_\delta (\xi ) = \xi ^\delta \)  where
\( \xi ^\delta = \pi _{+,*} \beta _{\rho ,\delta } (\xi _+ - \xi _+(w_+)) +\pi _{-,*} \beta _{\rho ,\delta } (\xi _- - \xi _-(w_-)) + \xi (w) \) 
for \( \kappa _\pm (z) \in B_{| \delta |^{1/2} \rho ^2} (0) \)  and \(\xi ^\delta = \xi \)  otherwise. Here \(x = \xi ^\pm (w^\pm )\)  is the value
of \(\xi \)  at the node. Define \( T_\delta := (I \times R_\delta )Q_\delta (I \times K_\delta ) .\)  That is, if \( (a,\xi ) = Q_\delta K_\delta (\phi ,\psi ,\eta ) \)  then \( T_\delta (\phi ,\psi ,\eta ) =(a,\xi ^\delta ) .\)  The map \(T_\delta \)  is the required approximate right
inverse. The difference \((\tilde{D}_\delta T_\delta -I)(\phi ,\psi ,\eta )\)  is the sum of terms
\( {\operatorname{d}}_{A,u^\delta } (a,\xi ^\delta ) - \phi , \quad {\operatorname{d}}_{A,u_0}^* (a,\xi ^\delta ) - \psi , \quad D_{A,u^\delta } (a,\xi ^\delta ) - \eta .\) 
By definition \({\operatorname{d}}_{A,u_0^\delta }(a,\xi ) = \phi \) , so the first
difference has contributions involving the difference between
\(u_0^\delta \)  and \(u^\delta \) , and between \(\xi ^\delta \)  and \(\xi \) .
But since \({\operatorname{d}}_{A,u}(a,\xi ) = {\operatorname{d}}_A a + u^* L_{\xi _X} P(\Phi )\) , these
terms are zeroth order in \(u,\xi \) ,
\( \Vert (u^\delta )^* L_{\xi ^\delta _X} P(\Phi ) - u_0^\delta L_{\xi _X}P(\Phi ) \Vert _{0,3} \le C | \delta | \Vert \xi \Vert _{C^0} ;\) 
that is, a constant times the area of \(\pi _{-,*} (\hat{\Sigma }_-^*)\) ,
which goes to zero as \(\delta \)  does. A similar discussion holds for
the second difference. The third difference has terms arising from
the cutoff function and the term \(a_X^{0,1}(u)\)  on the bubble region
\(|\kappa _+(z)| \le | \delta |^{1/2}\) .
We have
\( \Vert a_X^{0,1}(u) \Vert _{0,3,\delta } \le \Vert a_X^{0,1}(u) \Vert _{0,3} \le C |\delta | .\) 
since \(p \ge 2\) ; for \(p = 2\)  the \(W^{0,3,\delta }\)  and \(W^{0,3}\)  norms
are the same, by conformal invariance; for \(p > 2\)  the \(0,3\) -norm is
strictly greater. The term involving the derivative of the cutoff
function satisfies
\( \Vert {\operatorname{d}}\beta _{\rho ,\delta } (\xi - \xi (w)) \Vert _{0,3,\delta } \le c \log ( \rho ^2 )^{-2/3} \Vert (\xi - \xi (w)) \Vert _{1,3} \) 
by Lemma REF , and so vanishes in the limit \(\rho \rightarrow \infty \) . Using the uniform bound on \(Q_\delta \) , the total difference
is bounded by \(C ( \log ( \rho ^{2})^{-2/3} + |\delta |) \Vert (\phi ,\psi ,\eta ) \Vert \) , and so vanishes in the limit \(\delta \rightarrow 0,\rho \rightarrow \infty , |\delta | \rho ^2 \rightarrow 0\) . The uniform bound on \(T_{\delta }\) 
follows from the uniform bound on \(Q_\delta \)  and the cutoff and
extension operators.
Step 3: Uniform quadratic estimate
Proposition 4.2.8 
There exist constants \(c,C > 0\)  such that if \(\Vert \xi \Vert _{1,3,\delta } < c\) , \( |\delta | < c, \rho > 1/c\)  and \( |\delta |\rho ^4 < c \)  then the map \(\mathcal {F}^\delta _{A,{u}}\)  satisfies a
quadratic bound
\( \Vert D \mathcal {F}_{A,{u}}^\delta (a_1,\xi _1) -\tilde{D}_{A,{u}^\delta }(a_1,\xi _1) \Vert _{\delta } \le C \Vert a,\xi \Vert _{\delta } \Vert a_1, \xi _1 \Vert _{\delta } .\) 

The norm of the non-linear part of the curvature \(\Vert [a,a_1]\Vert _{{0,3}}\)  is bounded by Sobolev multiplication. The other term
appearing in the first vortex equation satisfies
\( \Vert \exp _{{u}^\delta } (\xi _0)^* L_{\xi _{1,X}}P(\Phi ) - ({u}^\delta )^* L_{\xi _{1,X}}P(\Phi ) \Vert _{{0,3}} \le C \Vert \xi _0 \Vert _{1,3,\delta } \Vert \xi _1 \Vert _{1,3,\delta } \) 
for some constant \(C\)  independent of \(\delta \) , using that
\(W^{1,3,\delta }\)  norm controls the \(W^{0,3}\)  norm uniformly. The
non-linear terms in the Cauchy-Riemann equation are estimated as in
Theorem REF  and [2]}; note
that we are fixing the complex structure on \(\Sigma \) , which avoids the
more complicated analysis we gave in the previous section. The second
vortex equation also involves a term of mixed type \( \Psi _{u}(\xi +\xi _1)^{-1} (a_1)_X^{0,1}(\exp _u(\xi + \xi _1)) - \Psi _u(\xi )^{-1}(a_1)_X^{0,1}(\exp _u(\xi )) .\)  It follows from uniform Sobolev
embedding that this difference has \(0,3,\delta \) -norm bounded by \(C\Vert a_1 \Vert _{{1,3}} \Vert \xi _1 \Vert _{1,3,\delta }\)  for some
constant \(C\)  independent of \(\delta \) .
Step 4: Implicit Function Theorem
Theorem 4.2.9  Let \((A,{u})\)  be a regular stable
nodal vortex of combinatorial type \(\Gamma \) . There exist constants
\(\epsilon _0,\epsilon _1 > 0\)  such that for every \((a,{\xi }, \delta ) \in \operatorname{Def}_\Gamma (A,{u})\)  with norm less than \(\epsilon _0\) , there exists
a unique \((\phi ,\psi ,\eta )\)  of norm less than \(\epsilon _1\)  such that if
\((a_1,\xi _1) = Q_\delta (\phi ,\psi ,\eta )\)  then \((A + a_0 + a_1,\exp _{{u}^{\delta }}({\xi }_0^{\delta } + \xi _1))\)  is a
symplectic vortex in Coulomb gauge with respect to
\((A,{u}^\delta )\) . The solution depends smoothly on
\(a_0,{\xi }_0\) , and transforms equivariantly the action of
\(\mathcal {G}(P)_{A,{u}}\)  on \(\operatorname{Def}_\Gamma (A,{u})\) .

Uniform error and quadratic estimates are those for
\(\mathcal {F}_{A,{u}}^\delta \)  in Lemmas REF , REF , and
REF , in a uniformly bounded neighborhood of 0 in
\(\operatorname{Def}_\Gamma (A,{u})\) . Then the first claim is an application of
the quantitative version of the implicit function theorem (see for
example [2]}). Equivariance follows from
uniqueness of the solution given by the implicit function theorem,
since the map \(\mathcal {F}^{D,\delta }_{A,{u}}\)  is equivariant for the
action of \(\mathcal {G}(P)_{A,{u}}\) .
Step 5: Rigidification
As in the case of holomorphic maps in the previous section, there is a
more natural way of parametrizing nearby symplectic vortices which
involves examining the intersections of the sections with submanifolds
of \(P(X)\) , and framings induced by parallel transport. First we study
the differentiability of the evaluation maps. The gluing construction
of the previous step gives rise to a deformation \((A_S,{u}_S)\)  of
\((A,{u})\)  with parameter space a neighborhood \(S\)  of 0 in
\(\operatorname{Def}(A,{u})\) , and so a map
\( S \rightarrow \overline{M}_n(\Sigma ,X), \ s \mapsto (\hat{\Sigma }_s,A_s,{u}_s) \) 
Consider the map
\( \operatorname{ev}: (\hat{\Sigma } - U) \times S \rightarrow P(X), \quad (z,s) \mapsto {u}_s(z) .\) 
Proposition 4.2.10 
The map \(\operatorname{ev}\)  of (REF ) is \(C^1\)  for the family constructed by
gluing in Theorem REF  using the exponential gluing profile.

We denote by \({u}_S^{\operatorname{pre}}: \hat{\Sigma }_S \rightarrow X\) 
the family obtained by pre-gluing only, that is, omitting the step
which solves for an exact solution. We denote by \(\operatorname{ev}^{\operatorname{pre}}\)  the map
\( \operatorname{ev}^{\operatorname{pre}}: (\hat{\Sigma } - U) \times S \rightarrow P(X), \quad (z,s) \mapsto {u}_s^{\operatorname{pre}}(z) .\) 
This map is independent of the gluing parameters, and is therefore
\(C^1\) . We write \(s = (a_0,\xi _0)\)  and \(A_s = A + a_0 + a_1, {u}_s= \exp _{{u}^{{\operatorname{pre}}}_s}(\xi _0^\delta + \xi _1)\) . The corrections
\(a_1,\xi _1\)  depend smoothly on \(a_0,\xi _0\) , by the implicit function
theorem, and so \(\xi _1(z)\)  depends smoothly on \(a_0,\xi _0\) . Next we
take the derivative with respect to the gluing parameter. Let
\((A,{u})\)  be a nodal symplectic vortex, \((A,{u}^\delta )\)  the
pre-glued pair (we omit the parameter \(\rho \)  controlling the diameter
of the gluing region from the notation) and consider the equation \(\mathcal {F}_{A,{u}^\delta } (a_0 + a_1, \xi _0^\delta + \xi _1) = 0 .\) 
Let \(\tilde{D}_\delta \)  denote the derivative of \(\mathcal {F}_{A,{u}^\delta }\) .
Differentiating with respect to \(\delta \)  gives
\( \tilde{D}_\delta \left(\frac{d}{d \delta }a_1 , D\exp _{{u}^\delta }(\xi _0^\delta ;0 , \frac{d}{d \delta }\xi _1 )\right) = - \tilde{D}_\delta \left(0,D\exp _{{u}^\delta }(\xi _0^\delta ; \frac{d}{d \delta }{u}^\delta , \frac{d}{d \delta }\xi _0^\delta )\right) .\) 
The same arguments as in the proof of Theorem REF  show that
there exists a constant \(C > 0 \)  such that the right hand side is
bounded in norm by \(C e^{-1/\delta }\) . On the other hand, the norm of
the left-hand side \(\tilde{D}_\delta \)  is uniformly bounded from below in
terms of the norm of \(\frac{d}{d \delta }a_1, \frac{d}{d \delta }\xi _1\) , by the quadratic
estimates. It follows that \((\frac{d}{d \delta }a_1, \frac{d}{d \delta }\xi _1)\)  is also bounded
in norm by \(C e^{-1/\delta }\) . Hence \( \lim _{\delta \rightarrow 0}\partial _\delta \operatorname{ev}= 0 \) . It follows that \(D\operatorname{ev}\)  has a continuous
limit as \({\delta } \rightarrow 0\) .
Choose a path \(\gamma : [0,1] \rightarrow \Sigma \)  in the principal component
and an element \(\phi _0 \in P_{\gamma (0)}\) . Let \(\tau _\gamma (A):P_{\gamma (0)} \rightarrow P_{\gamma (1)}\)  denote parallel transport. By an
\(m\) -framed family of marked curves, we mean a family of curves
together with an \(m\) -tuple of points in \(P\) . Given a family
\((\hat{\Sigma }_S,A_S,{u}_S)\)  of gauged holomorphic maps over a
parameter space \(S\) , a collection of codimension two submanifolds
\({Y} = (Y_1,\ldots ,Y_k)\)  in \(P(X)\) , and a collection of paths
\({\gamma } = (\gamma _1,\ldots , \gamma _l)\)  with the same initial
point \(y_0\)  to \(y_j, j = 1,\ldots , l\) , define a family of marked,
framed curves
\(\hat{\Sigma }_S^{{Y},{u},{\gamma },A} \rightarrow S\) 
by requiring that the additional marked points \(z_{n+i}\)  map to \(Y_i\) ,
and the framings are given by parallel transport along the paths
\(\gamma _i\) .
Definition 4.2.11 
The data \({Y},{\gamma },A,{u}\)  are compatible if

each \(Y_j\)  intersects \(u_j\)  transversally in a single point \(z_j\in \hat{\Sigma }\) ;

if \((a,\xi ) \in  \operatorname{ker}\tilde{D}_{A,{u}}\)  satisfies \(\xi (z_{n+j})\in T_{{u}(z_{n+j})} P(X) \)  for \(j = 1,\ldots , k\)  and \(D_A\tau _{\gamma _i}(a) = 0 \)  for \(i = 1,\ldots , l\)  then \((a,\xi ) = 0\) .

the curve \(\hat{\Sigma }\)  marked with the additional points
\(z_{n+1},\ldots , z_{n+k}\)  is stable.

if some automorphism of \((\hat{\Sigma },{u})\)  maps \(z_i\)  to
\(z_j\)  then \(Y_i\)  is equal to \(Y_j\) .


The second condition says that there are no infinitesimal deformations
which do not change the positions of the extra markings or framings.
Proposition 4.2.12 
Let \((A,{u})\)  be a parametrized regular stable nodal vortex, and
\((A_S, {u}_S) \rightarrow S\)  the stratified-smooth universal deformation
constructed by the gluing construction. There exists a collection
\(({Y},{\gamma })\)  compatible with \(({u},A)\) . Furthermore, if
\(({Y},{\gamma })\)  is compatible with \((A,{u})\) , then the
rigidified family \(\hat{\Sigma }_S^{{Y},{u},{\gamma },A} \rightarrow S\)  of (REF ) is a stratified-smooth deformation of the
marked-curve-with-framings
\(\hat{\Sigma }^{{Y},{u},{\gamma },A}\)  which defines an
immersion of \(S\)  into the parameter space for the universal
deformation of the central fiber.

First we show the existence of a compatible collection. Suppose that
the second condition is not satisfied for some \((a,\xi )\) . Suppose
first that \(\xi \ne 0\) . Let \(z_{n+1}\)  be a point with \(\xi (z_{n+1})\ne 0\) , and choose a codimension two submanifold \(Y_{n+1}\)  transverse
to \(u\)  near \(u(z_{n+1})\) , and such that \(TY_{n+1}\)  does not contain
\(\xi (z_{n+1})\) . Adding \(Y_{n+1}\)  to the list of submanifolds
decreases the dimension of the space of \((a,\xi )\)  satisfying the
condition in (b) by at least one. Repeating this process, we may
assume that the only elements satisfying the condition in (b) have
\(\xi =0 \) . Suppose that \(\xi \)  is zero, so that \(a\)  is necessarily
non-zero. Choose an additional marked point \(y_{l+1}\)  and a path
\(\gamma _{l+1}\)  from the base point \(y_0\)  to \(y_{l+1}\)  such that the
derivative of the parallel transport over \(\gamma \)  with respect to \(a\) 
over is non-zero. Appending \(\gamma _{l+1}\)  to the list of path
decreases the dimension of \((a,\xi )\)  satisfying the condition in (b)
by at least one. Hence the process stops after finitely many steps,
after which the kernel is trivial. The proof of the second claim is
similar to Proposition REF  and will be omitted.
Step 6: Surjectivity
We show that any nearby vortex appears in the family constructed
above. First, we show:
Proposition 4.2.13 
Let \((A,{u})\)  be a regular strongly stable symplectic vortex.
There exists a constant \(\epsilon > 0\)  such that if \((A_1,u_1) = (A + a,\exp _{{u}^\delta }(\xi ))\)  with \(\Vert a \Vert _{1,3} + \Vert \xi \Vert _{1,3,\delta } \le \epsilon \)  then after gauge transformation we have
\((A_1,u_1) = (A+ a_0 + a_1,\exp _{{u}^\delta }(\xi _0^\delta +\xi _1))\)  for some \((a_0,\xi _0) \in  \operatorname{ker}(\tilde{D}_{A,{u}})\)  and
\((a_1,\xi _1)\)  in the image of \(Q_\delta \) .

We claim that for some constant \(C > 0\) , we have \((a,\xi ) =(a_0,\xi _0^{{\delta }}) + (a_1,\xi _1)\)  for some \((a_0,\xi _0) \in  \operatorname{ker}\tilde{D}_{A,{u}}\)  and \((a_1,\xi _1) \in \operatorname{Im}\tilde{D}_{A,{u}^{{\delta }}}^*\)  with norm \( \Vert (a_1,\xi _1) \Vert \le C \Vert (a_1,\xi _1) \Vert .\)  Given the claim, the proposition
follows by the uniqueness statement of the implicit function theorem.
For any \(c > 0\)  there exists \(\delta _0\)  such that for \(\delta <\delta _0\) ,
\( \Vert \tilde{D}_{A,{u}^{{\delta }}} (a_0^{{\delta }},{\xi }_0^{{\delta }}) \Vert \le c \Vert (a_0^{{\delta }},{\xi }_0^{{\delta }}) \Vert \) 
by estimates similar to those of Lemma REF . Thus the image
of \( \operatorname{ker}\tilde{D}_{A,{u}}\)  is transverse to \(\operatorname{Im}\tilde{D}_{A,{u}^\delta }^*\) , for \(\delta \)  sufficiently small, since
it meets \(\operatorname{Im}\tilde{D}_{A,{u}^\delta }^*\)  trivially and projects
isomorphically onto \( \operatorname{ker}\tilde{D}_{A,{u}^\delta }\) , by gluing for
indices, as in [18]}. The claim then follows
from the inverse function theorem.
Given a strongly stable symplectic vortex \((A,{u})\)  with stable
domain \(\hat{\Sigma }\) , let \((A_S,{u}_S)\)  be the family given by
the gluing construction above. Otherwise, if \({Y}\)  is not stable,
let \({Y} = (Y_1,\ldots , Y_l)\)  be a collection of codimension two
submanifolds of \(P(X)\) , and consider the family
\((A,{u}^{{Y}})\)  with additional marked points given by
requiring that the additional marked points \(z_{n+i}\)  map to \(Y_i\) .
Let \((A_S,{u}_S)\)  denote the family obtained by applying the
gluing construction for \((A_S,{u}_S^{{Y}})\) , and then
forgetting the additional marked points.
Lemma 4.2.14 Suppose that \((A_i,u_i)\)  Gromov converges to \((A,u)\) .
After a sequence of gauge transformations, for any \(\epsilon \) , there
exists \(i_0\)  such that if \(i > i_0\)  then there exists
\(\delta ,(a_i,\xi _i)\)  satisfying \((A_i,u_i) = (A + a_i,\exp _{{u}^\delta }(\xi _i))\)  with \(\Vert a_i \Vert _{1,3} + \Vert \xi _i \Vert _{1,3,\delta } \le \epsilon \) .

By definition of Gromov convergence, after gauge transformation \(A_i\)  \(C^0\) -converges to \(A\) 
and converges uniformly in all derivatives on the complement of the
bubbling set [33]}. The exponential decay estimate
[33]} show that \(u_i\)  converges to \(u\)  on the
complement of the nodes, uniformly in all derivatives on compact sets,
and whose derivative on the gluing region is uniformly bounded in the
\(\delta \) -dependent metric. It follows that \(u_i =\exp _{u^\delta }(\xi _i)\)  for some \(\delta \)  and \(\xi _i \in \Omega ^0(\Sigma ^\delta , (u^\delta ) T^{\operatorname{vert}} P(X))\)  with \(\Vert \xi _i \Vert _{1,3,\delta } < \epsilon \) . To obtain the improved convergence
for the connection, note that \( F_{A_i} + (u_i)^* P(\Phi ) = 0 \)  and
the corresponding equations for the limit \((A,u)\)  imply that
\( F_{A_i} - F_A = {\operatorname{d}}_A (A_i - A) - (A - A_i) \wedge (A - A_i) =(u_i)^* P(\Phi ) - u^* P(\Phi ) .\) 
Since \(u_i^* P(\Phi )\)  is bounded and converges to \(u^* P(\Phi )\)  on the
complement of the bubbling set, and \(A_i\)  converges to \(A\)  in \(C^0\) 
hence \(W^{0,3}\) , the right hand side converges to 0 in \(W^{0,3}\)  as
\(i \rightarrow \infty .\)  After gauge transformation we may assume that \({\operatorname{d}}_A^* (A - A_i) = 0\) . Then the elliptic estimate for the operator
\({\operatorname{d}}_A \oplus {\operatorname{d}}_A^*\)  implies that \(A - A_i\)  converges to zero in
\(W^{1,3}\) .
Corollary 4.2.15 
\((A_S,{u}_S)\)  is a stratified-smooth versal deformation of
\((\hat{\Sigma },A,{u})\) .

Proposition REF  implies that any family
\((A_{S^1}^1,{u}_{S^1}^1)\)  is obtained by pull-back from
\((A_S,{u}_S)\) , in case \(\hat{\Sigma }\)  is stable, or obtained from
the family obtained by adding the marked points mapping to
submanifolds, in general.
Step 7: Injectivity
We show that any nearby vortex appears once in our family, up to the
action of \( \operatorname{Aut} (A,u)\) ; this is part of the following:
Theorem 4.2.16 Any family \((A_S,{u}_S)\)  constructed
by gluing using the exponential gluing profile is a strongly universal
stratified-smooth deformation of \((A,{u})\) .

Let \(\overline{Z}_n(P,X)\)  denote the moduli space of marked symplectic
vortices up to equivalences that involve only the identity gauge
transformation, so that \( \overline{M}_n(P,X) = \overline{Z}_n(P,X) / \mathcal {G}(P) .\)  Let
\((A,{u})\)  be a stable marked vortex, and \(W_{A,{u}}\)  a slice
for the gauge group action on \(\overline{Z}(P,X)\) , so that
\(W_{A,{u}}/\mathcal {G}(P)_{A,{u}} \rightarrow \overline{M}_n(P,X) \) 
is a homeomorphism onto its image. Let \( \operatorname{Aut} _0(A,{u})\)  denote the
subgroup of \( \operatorname{Aut} (A,u)\)  acting trivially on \(P\) , so that
\(\mathcal {G}(P)_{A,{u}} =  \operatorname{Aut} (A,u)/ \operatorname{Aut} _0(A,{u})\)  is the stabilizer of
\((A,{u})\)  under the gauge action. Let \((A_S,{u}_S)\)  denote a
universal deformation of \((A,{u})\)  constructed by gluing using the
exponential gluing profile. We claim that the map
\(  S/ \operatorname{Aut} _0(A,{u}) \rightarrow W_{A,{u}}, \quad [s]\mapsto [A_s,{u}_s] \) 
is an injection. Indeed, rigidification produces an injection
\(  S/ \operatorname{Aut} _0(A,{u})\rightarrow \overline{M}_{n+k,l}(\Sigma )/ \operatorname{Aut} _0(A,{u}), \quad [s] \mapsto [\Sigma ^{A_s,{u}_s,{Y},{\gamma }}] \) 
where \( \operatorname{Aut} _0(A,{u})\)  acts by re-ordering the marked points.
Since this map factors through (REF ), the claim follows. If
\((A_{S^1},{u}_{S^1}^1)\)  is a family of symplectic vortices giving
a deformation of any fiber of \((A_S,{u}_S)\) < then Corollary
REF  together with injectivity shows that this family is
obtained by pull-back by some map \(S^1 \rightarrow S\) . Hence
\((A_S,{u}_S)\)  is a stratified-smooth strongly universal
deformation of \((A,{u})\) .
Theorem 4.2.17  Let \(X\)  be a Hamiltonian \(G\) -manifold
equipped with a compatible invariant almost complex structure \(J \in J(X)^G\) . The maps
\( S \rightarrow \overline{M}_n(\Sigma ,X), \quad s\mapsto [A_s,{u}_s] \) 
associated to the universal deformations constructed above equip the
locus \(\overline{M}^{\operatorname{reg}}_n(\Sigma ,X)\)  of regular stable symplectic vortices
with the structure of a stratified-smooth orbifold. If the local
coordinates near the nodes are chosen compatibly and the gluing
profile is the exponential gluing profile, then the deformations
provide \(\overline{M}^{\operatorname{reg}}_n(\Sigma ,X)\)  with the structure of a
\(C^1\) -orbifold.

It suffices to show that the charts given by two sets
\({Y}_j,{\gamma }_j\)  are compatible. Define \({Y} ={Y}_1 \cup {Y}_2\)  and \(m = m_1 + m_2\)  the total number of
extra points. Similarly let \({\gamma }\)  be the union of
\({\gamma }_1\)  and \({\gamma _2}\)  of total number \(l = l_1 +l_2\) . The family \( \hat{\Sigma }^{{Y},{u},{\gamma },A}_S\) 
admits a proper étale forgetful map \(\hat{\Sigma }_S^{{Y},{u},{\gamma },A} \rightarrow \hat{\Sigma }_S^{{Y}_j,{u},{\gamma }_{j},A}, \quad j = 1,2\) 
whose fiber consists of the re-orderings of the points for \({Y}\) 
induced by automorphisms of \( \operatorname{Aut} (A,{u})\)  that fix the ordering
for \({Y}_j\) . It follows that the corresponding charts are
\(C^1\) -compatible.
Remark 4.2.18 As discussed in Remark REF , the Theorem
implies that if \(\overline{M}^{{\operatorname{reg}}}_n(\Sigma ,X)\)  is compact then it admits
a (non-canonical) smooth structure.

Let \(\overline{M}_n(\Sigma )\)  denote the moduli space of stable maps to
\(\Sigma \)  with homology class \([\Sigma ]\) , \(n\)  markings and genus that
of \(\Sigma \) , or in other words, parametrized stable curves with
principal component isomorphic to \(\Sigma \) . Forgetting the pair
\((A,u)\)  gives a forgetful morphism \( \overline{M}^{{\operatorname{reg}}}_n(\Sigma ,X) \rightarrow \overline{M}_n(\Sigma ) .\)  Using the differentiable structure defined above,
the evaluation maps are differentiable but unfortunately the
forgetful morphisms are not, unless one uses a different gluing
profile for the moduli space of vortices with one less marking. More
precisely, the forgetful morphism \(\overline{M}^{\operatorname{reg}}_n(\Sigma ,X) \rightarrow \overline{M}_n(\Sigma )\)  is continuous and \(C^1\)  near any pair \((A,{u})\) 
whose domain is stable as an element of \(\overline{M}_n(\Sigma )\) , and a
submersion near the boundary of \(\overline{M}_n(\Sigma )\) . For the standard
smooth structure on \(\overline{M}_n(\Sigma )\) , the forgetful morphism
\(\overline{M}^{\operatorname{reg}}_n(\Sigma ,X) \rightarrow \overline{M}_n(\Sigma )\)  is smooth.
The gluing construction has various parametrized versions. For
example, in [35]} we consider a moduli space of polystable
polarized vortices, which consist of a vortex together with a lift
of the connection to the Chern-Simons line bundle. In each of these
cases one applies the implicit function theorem using the linearized
operator for the parametrized problem to prove that any parametrized regular polystable vortex has a strongly universal
deformation in the parametrized sense. In particular, any regular
polystable polarized vortex has a strongly universal deformation etc.
